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MODULATION CALORIMETRY AND RELATED TECHNIQUES
Yaakov KRAFTMAKHER Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel IPCMS - CNRS (UM 7504), 23 rue du loess, F-67037 Strasbourg Cedex, France
AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO
Physics Reports 356 (2002) 1–117
Modulation calorimetry and related techniques Yaakov Kraftmakher∗ Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Received November 2000; editor: A:A: Maradudin Contents 1. Introduction 2. Theory 2.1. Basic equation of modulation calorimetry 2.2. Temperature oscillations in resistively heated wires 2.3. Thermal coupling and thermal di0usivity of the sample 3. Modulation of heating power 3.1. Direct electric heating 3.2. Modulated-light heating 3.3 Other methods 4. Measurement of temperature oscillations 4.1. Use of oscillations in the sample’s resistance 4.2. Photoelectric detectors 4.3. Thermocouples and resistance thermometers 4.4. Lock-in detection of periodic signals 5. Modulation calorimetry 5.1. Methods of calorimetry, a brief review 5.2. Modulation calorimetry at high temperatures 5.3. Low temperatures 5.4. Measurements under high pressures 5.5. Modulation microcalorimetry
3 5 5 8 9 11 12 12 15 16 17 22 24 25 26 26 32 40 42 44
5.6. Photoacoustic technique 5.7. Noncontact calorimetry 5.8. Modulated di0erential scanning calorimetry 5.9. Commercial modulation calorimeter ACC-1 6. Modulation dilatometry 6.1. Methods of dilatometry, a brief review 6.2. Principle of modulation dilatometry, wire samples 6.3. Di0erential method 6.4. Bulk samples 6.5. Interferometric modulation dilatometer 6.6. Nonconducting materials 6.7. Measurement of extremely small periodic displacements 7. Modulation measurements of electrical resistance, thermopower, and spectral absorptance 7.1. Temperature derivative of resistance 7.2. Direct measurement of thermopower 7.3. Spectral absorptance 8. Noise thermometry of wire samples 8.1. Principles of noise thermometry 8.2. Noise correlation thermometer
∗
Corresponding author. Fax: +972-3-535-32-98. E-mail address:
[email protected] (Y. Kraftmakher).
c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 3 1 - X
48 49 49 50 52 52 55 57 58 60 61 62 63 63 65 66 67 67 68
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8.3. Determination of the temperature derivative of the resistance 9. Electronic instrumentation for modulation measurements 10. Accuracy of modulation measurements 11. Applications of modulation techniques 11.1. Equilibrium point defects in metals 11.2. Phase transitions in solids
69 70 73 75 75 80
11.3. Temperature Euctuations and isochoric speciFc heat of solids 11.4. Relaxation phenomena in the speciFc heat 12. Conclusion Acknowledgements References
92 95 104 105 105
Abstract Modulation techniques for measuring speciFc heat, thermal expansivity, temperature derivative of resistance, thermopower, and spectral absorptance are reviewed. Owing to the periodic nature of temperature oscillations, high sensitivity and excellent temperature resolution are peculiar to all these methods. Various methods of modulation and of measuring the temperature oscillations are presented. Some applications of the modulation techniques for studying physical phenomena in solids and liquids are considered (equilibc 2002 Elsevier Science rium point defects, phase transitions, relaxation phenomena in speciFc heat). B.V. All rights reserved. PACS: 07.20.−n; 65. Keywords: Calorimetry; Modulation techniques; Thermophysical properties; Relaxation phenomena
Y. Kraftmakher / Physics Reports 356 (2002) 1–117
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1. Introduction Modulation techniques of studying thermophysical properties consist in periodically modulating the power that heats the sample and creating thereby temperature oscillations in the sample around a mean temperature. The amplitude of these oscillations depends on the heat capacity of the sample. Corbino (1910) discovered this principle at the beginning of the last century. However, at that time such measurements could not be precise. In modulation calorimetry, it is enough to measure the oscillations of the heating power and of the sample’s temperature. The use of periodic temperature changes provides important advantages. When the modulation frequency is suIciently high, corrections for heat losses from the sample are negligible even at the highest temperatures. In this respect, the method is comparable to adiabatic calorimetry. Against the pulse method, the modulation technique has the advantage that selective ampliFers and lock-in detectors measure the harmonic temperature oscillations. This feature becomes very important when good temperature resolution is required as, e.g., in studies of phase transitions. The modulation methods provide a unique possibility to perform measurements with temperature oscillations in the range 0.1–10 mK and even smaller if necessary. ModiFcations of the method di0er in the ways of modulating the heating power (heating by an electric current, radiation or electron-bombardment heating, induction heating, use of separate heaters, Peltier heating) and by the methods of detecting the temperature oscillations (by the resistance of the sample or radiation from it, and by the use of thermocouples, resistance thermometers, or pyroelectric sensors). The modulation techniques allow one to perform measurements in a wide temperature range, from fractions of a kelvin up to melting points of refractory metals, and with high sensitivity. In many cases, it is possible to assemble compensation schemes whose balance does not depend on the amplitude of the oscillations in the applied power and to automatically record quantities to be measured. The measurements can be controlled by a data-acquisition system and fully automated. All these features have made the method very attractive and widely used. In treating the data, the mean temperature and the amplitude of the temperature oscillations are considered constant throughout the sample. In this respect, the modulation techniques di0er from the method of temperature waves. As a rule, the measurements are carried out in a regime where the amplitude of the temperature oscillations in the sample is inversely proportional to its heat capacity. Corbino (1910) developed the theory of modulation calorimetry and carried out the Frst modulation measurements of speciFc heat. He used the oscillations of the sample’s resistance to determine the temperature oscillations. They were detected by passing through the sample a supplementary AC current of a frequency equal to that of the temperature oscillations or by the third-harmonic technique (Corbino, 1911). Fermi (1937) highly praised this work in a paper in memory of Corbino entitled “Un Maestro”. In some measurements, the temperature oscillations were detected by the oscillations of the thermionic current from the sample (Smith and Bigler, 1922; Bockstahler, 1925). Zwikker (1928) measured the speciFc heat of tungsten up to 2600 K. Considerable progress in modulation calorimetry has been achieved in the 1960s due to advances in experimental techniques. At the Frst stage, the method was used exclusively at high temperatures. Its most important feature was the smallness of the correction for heat losses. The samples, in the form of a wire or a rod, were heated by an electric current passing
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Y. Kraftmakher / Physics Reports 356 (2002) 1–117
through them. The temperature oscillations were deduced from oscillations in the resistance of the samples or radiation from them. By the use of this method, the high-temperature speciFc heat of refractory metals has been determined. Later, the method was employed in studies of phase transitions, where the main requirement became good temperature resolution. In these experiments, the samples were also heated by an electric current. At the second stage, the modulation technique was applied to measurements at low and middle temperatures and for studying nonconducting materials. Even at low and middle temperatures, the traditional domain of adiabatic calorimetry, modulation calorimetry ensures better temperature resolution and higher sensitivity. In addition, small dimensions of the samples often are of importance. Absolute values obtainable by the modulation technique are less accurate than those obtained by adiabatic calorimetry. Even under very favorable conditions, the accuracy of modulation measurements is about 1–2%. Nevertheless, the method is preferable in studies of phase transitions due to its excellent resolution. It is applicable under high pressures. Filippov (1960) was the Frst to employ the third-harmonic method for studying thermal properties of a liquid surrounding a probe. Temperature oscillations in a heater immersed in a liquid under study depend on the product of the speciFc heat and thermal conductivity of the liquid. This method was rediscovered by Birge and Nagel (1985) and used in their well-known studies of supercooled liquids near the glass transition. The authors founded a new branch in modulation calorimetry, speciFc-heat spectroscopy. For a long time, there was no special term for modulation calorimetry. The term ‘modulation method for measuring speciFc heat’ was proposed in the paper describing the equivalentimpedance method for determining the speciFc heat of wire samples (Kraftmakher, 1962). However, most investigators have become acquainted with modulation calorimetry only from the famous papers of Sullivan and Seidel (1966, 1967, 1968). These authors considered thermal coupling in a system including a sample, a heater, and a thermometer, and performed measurements at low temperatures. Sullivan and Seidel (1968) stressed the signiFcant advantages of this technique as follows. “(1) The sample may be coupled thermally to a bath. (2) The method is a steady-state measurement. (3) Changes in heat capacity with some experimentally variable parameter may be recorded directly. (4) Extremely small heat capacities may be measured with accuracy. (5) The method possesses a precision an order of magnitude better than existing techniques.” The term ‘AC calorimetry’ introduced by the authors is now generally accepted. A new term recently appeared, ‘the temperature-modulated calorimetry’ (Gmelin, 1997). The excellent resolution peculiar to modulation calorimetry appeared very useful in studies of high-temperature superconductors. The aim was to investigate contributions that amount only to a few percent of the total speciFc heat. Small samples are generally used in modulation calorimetry, but a challenge has arisen when it was necessary to perform measurements on a microgram sample. Modulation calorimetry provides unique opportunities: temperatures down to 0:1 K (Feng et al., 1988; Steinmetz et al., 1989); magnetic Felds up to 30 T (Yu et al., 1988; Fortune et al., 1990); pressures up to 3:5 GPa (Eichler and Gey, 1979; Eichler et al., 1980, 1981); samples as small as 1 g (Fominaya et al., 1997a,b, 1999a,b); a resolution of the order of 0.01%. Temperature oscillations necessary for calorimetric measurements are of the order of 1 K at high temperatures, of 1 mK at room temperatures, and of 1 K at liquid helium temperatures (Mehta and Gasparini, 1997, 1998; Mehta et al., 1999). One can measure speciFc
Y. Kraftmakher / Physics Reports 356 (2002) 1–117
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heat as a function of an external parameter, e.g., magnetic Feld or pressure. Sullivan and Seidel (1967, 1968) have proposed and conFrmed this approach. One of the recent achievements is the noncontact calorimetry successfully employed in space during a mission of the shuttle Columbia (Wunderlich et al., 1997). Modulation calorimetry became such a necessary technique that a commercial instrument has already appeared (calorimeter ACC-1, Sinku-Riko, Inc.). The modulation principle is also powerful for studying some other thermophysical properties. Measurements of oscillations in the sample’s length permit a direct determination of thermal expansivity. This technique enables one to avoid the main drawback of high-temperature dilatometry caused by the creep of the samples and to signiFcantly improve the temperature resolution. The temperature derivative of resistance is available by registering oscillations of the sample’s resistance caused by the temperature oscillations. Direct measurements of this quantity more reliably reveal the behavior of electrical resistivity, e.g., near phase transitions of the second order. Measuring temperature oscillations by two thermocouples provides a direct comparison of their thermopowers. The modulation method was applied also to measurements of the spectral absorptance. High sensitivity and unique temperature resolution are peculiar to all modulation techniques. Both features are due to the periodic nature of the temperature changes. Employment of selective ampliFers and lock-in detectors reduces any inEuence of noise and interference. The theory of the measurements is simple and quite adequate to experimental conditions. In contrast to pulse or dynamic calorimetry, the modulation technique is a steady-state method: the amplitude and the phase of the temperature oscillations in the sample do not depend on time. Electronic equipment necessary for modulation measurements is widely used in fundamental and applied studies and is quite accessible. It is therefore possible to perform the measurements using common scientiFc instruments, as well as data-acquisition systems and computers for controlling the measurements and processing the data. Until today, only modulation calorimetry gained recognition by the scientiFc community. Other modulation methods are still waiting for a wider practical use. Some modulation techniques were reinvented several times. This fact is a convincing conFrmation of their usefulness. On the other hand, the principle of modulation was discovered long ago, and its application to studying various physical properties seems to be natural and even ordinary. The long history of the modulation techniques is presented in Table 1.1. Many review papers and book chapters are devoted to the modulation calorimetry (Filippov, 1966, 1967, 1984; Gmelin, 1997; Kraftmakher, 1973a, 1984, 1988, 1992a). SpeciFc items of modulation calorimetry were considered by Hatta and Ikushima (1981), Garland (1985), Huang and Stoebe (1993), Finotello and Iannacchione (1995), Birge et al. (1997), Finotello et al. (1997), Hatta (1997a, b), Jeong (1997), Minakov (1997), Hatta and Nakayama (1998), Hatta and Minakov (1999), Kraftmakher (1992b, 1994a, 1996a), and Wunderlich (2000). The author has reviewed modulation dilatometry and other modulation techniques (Kraftmakher 1973b, 1978a, 1989). 2. Theory 2.1. Basic equation of modulation calorimetry The basic theory of modulation calorimetry is very simple (Corbino, 1910). The power heating the sample is modulated by a sine wave and thus equals p0 +p sin !t. The sample’s temperature
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Table 1.1 Brief presentation of long history of modulation calorimetry and related techniques Item
Reference
Theory, supplementary-current and 3! methods SpeciFc heat of tungsten up to 2600 K Probe inside a medium, 3! technique Equivalent-impedance method SpeciFc heat of tungsten, 1500 –3600 K Photoelectric detectors Molten metals, high temperatures Electron-bombardment heating Modulation dilatometry Analysis of resistively heated wires Measurement at various frequencies Account of thermal coupling, low temperatures Modulated-light heating Modulation measurement of dR=dT Induction heating Nonconducting materials Modulation measurement of thermopower
Corbino (1910, 1911) Zwikker (1928) Filippov (1960), Rosenthal (1961, 1965) Kraftmakher (1962) Kraftmakher and Strelkov (1962) Lowenthal (1963) Akhmatova (1965, 1967) Filippov and Yurchak (1965) Kraftmakher and Cheremisina (1965) Holland and Smith (1966) Smith (1966) Sullivan and Seidel (1966, 1967, 1968) Handler, Mapother and Rayl (1967) Kraftmakher (1967a), Salamon et al. (1969) Filippov and Makarenko (1968) Glass (1968) Freeman and Bass (1970), Hellenthal and Ostholt (1970), Kraftmakher and Pinegina (1970) Zally and Mochel (1971) Manuel and VeyssiRe (1972, 1973) Chu and Knapp (1973) Varchenko and Kraftmakher (1973) Filippov et al. (1976) Schantz and Johnson (1978) Smaardyk and Mochel (1978) Tanasijczuk and Oja (1978) Kraftmakher and Cherepanov (1978) Kraftmakher and Krylov (1980a, b) Kraftmakher (1981) Birge and Nagel (1985, 1987) Pitchford et al. (1986) Johansen (1987) Graebner (1989) Monazam et al. (1989) Gill et al. (1993), Reading et al. (1993) Fominaya et al. (1997a, b); Riou et al. (1997) Wunderlich et al. (1997), Egry (2000)
Microcalorimetry, low temperatures Temperatures down to 0:3 K High pressures, low temperatures Nonadiabatic regime High pressures, high temperatures Liquid crystals Organic liquids Organic and biological materials Active thermal shield Observation of temperature Euctuations Modulation frequencies up to 105 Hz SpeciFc-heat spectroscopy, 0.01–6000 Hz Free-standing thin Flms Dilatometry of nonconducting materials Modulated-bath calorimetry Noncontact calorimetry Modulated di0erential scanning calorimetry Nanocalorimetry, low temperatures Noncontact calorimetry in space
therefore oscillates around a mean value T0 . For a short time interval Tt, during which the quantities involved remain constant, the heat-balance equation takes the form (p0 + p sin !t)Tt = mcTT + P(T )Tt :
(2.1)
Here m; c and T are the mass, speciFc heat and temperature of the sample, P(T ) is the power of the heat losses from the sample, ! = 2 f is the angular modulation frequency, and TT is
Y. Kraftmakher / Physics Reports 356 (2002) 1–117
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Fig. 2.1. Frequency dependence of the amplitude and phase of temperature oscillations and polar diagram 0 () (Rosenthal, 1965). !0 is the frequency for which tan = 1.
a change of the temperature during the interval Tt. The equation has a very simple meaning: heat input = heat accumulated in the sample + heat losses. Assuming T = T0 + ; T0 , and taking P(T ) = P(T0 ) + P (P = dP=dT is called the heat transfer coeIcient), one obtains mc + P(T0 ) + P = p0 + p sin !t ;
(2.2)
where = d=dt. The steady-state solution to this equation is P(T0 ) = p0 ;
(2.3a)
= 0 sin(!t − ) ;
(2.3b)
0 = (p=mc!) sin = (p cos )=P = p=(m2 c2 !2 + P 2 )1=2 ;
(2.3c)
tan = mc!=P :
(2.3d)
These results may be presented as a frequency dependence of 0 and tan , as well as a polar diagram 0 () (Fig. 2.1). An analogy exists between a sample subjected to modulated heat input and an integrating RC circuit fed by a voltage containing both DC and AC components. ∼ 1) is a criterion of the so-called adiabatic regime. More rigorThe condition tan 1 (sin = ◦ ously, it should be called quasiadiabatic. When the phase shift is close to 90 , the correction for heat losses is insigniFcant. The adiabaticity condition means that the oscillations of the heat losses due to the temperature oscillations in the sample are much smaller than the oscillations of the heating power. Although the heat transfer coeIcient grows rapidly with temperature, the regime of the measurements can be kept adiabatic by increasing the modulation frequency. Under adiabatic conditions, the heat capacity of the sample obeys the relation mc = p=!0 ;
(2.4)
which is the basic equation of modulation calorimetry. The temperature oscillations may be written in a complex form showing both the amplitude and the phase of the temperature oscillations: = p=(P + imc!) :
(2.5)
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Under adiabatic conditions, when P mc!, = −ip=mc! :
(2.5a)
In a nonadiabatic regime, when tan ¡ 10, the heat capacity obeys the relation mc = (p=!0 ) sin :
(2.6)
2.2. Temperature oscillations in resistively heated wires When an electric current heats a wire sample, it is necessary to take into account the temperature dependence of the sample’s resistance as was done by Corbino (1910). Hence, R=R0 +R , where R0 and R are the sample’s resistance and its temperature derivative at the mean temperature. When a DC current I0 superimposed with a small AC component i sin !t passes through the sample, the electric power dissipated in it equals I02 R0 + I02 R + 2I0 iR0 sin !t + 2I0 iR sin !t + i2 R0 sin2 !t + i2 R sin2 !t :
(2.7)
The last three terms are negligible because iI0 and R R0 . In addition, these terms correspond to power oscillations of higher frequencies, while selective ampliFers tuned to the fundamental frequency and=or lock-in ampliFers are usually employed in the measurements. The heat-balance equation for the oscillating part of the heating power is given by mc + (P − I02 R ) = 2I0 iR0 sin !t ;
(2.8)
and the solution to it di0ers from the expressions (2:3) only by a phase shift: = 0 sin(!t − ’) ;
(2.9a)
0 = (2I0 iR0 sin ’)=mc! ;
(2.9b)
tan ’ = mc!=(P − I02 R ) :
(2.9c)
Here ’ is the phase shift between the AC component of the heating current and the temperature oscillations. It was assumed that the current passing through the sample does not depend on the sample’s resistance because of the high internal resistance of the source. When the internal resistance of the source is negligibly small, then = 0 sin(!t − ) ;
(2.10a)
0 = (2U0 U sin )=R0 mc! ;
(2.10b)
tan
= mc!=(P + I02 R ) ;
(2.10c)
where U0 and U are the DC and AC components of the voltage applied to the sample, and is the phase di0erence between the AC component and the temperature oscillations. Usually, I02 R is several times smaller than P . The di0erence between the phase shifts ; ’ and is of no importance when the measurements are performed in the adiabatic regime when mc!P . In this case, there is no need to take into account the temperature dependence of the sample’s resistance. The di0erence becomes important only in a nonadiabatic regime.
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Fig. 2.2. Presentation of calorimetric cell consisting of a heater, a sample, and a thermometer. As a rule, the modulation period is much longer than the equilibration time inside the cell and much shorter than that between the sample and the bath (Sullivan and Seidel, 1968).
Under direct electrical heating, axial and radial temperature gradients may appear in the samples. However, Holland and Smith (1966) have shown that for thin samples and under proper modulation frequencies the gradients are suIciently small. The authors gave expressions for the magnitude and phase of all harmonic components of the temperature oscillations. Radial and end e0ects were treated as corrections to the solutions for a long thin wire. They may be required when the temperature coeIcient of the sample’s resistance is very large. Second, the surface temperature oscillations may di0er from those in the interior. Fortunately, such corrections can become meaningful only in special, rather exotic cases. 2.3. Thermal coupling and thermal di5usivity of the sample When using separate heaters and thermometers, one has to take into account thermal coupling in a calorimetric cell, as well as the Fnite thermal conductivity of the sample. Sullivan and Seidel (1968) considered a heater (heat capacity Ch , temperature Th ), a sample (Cs ; Ts ), and a thermometer (Ct ; Tt ) interconnected by thermal conductances Kh and Kt (Fig. 2.2). AC power applied to the heater provides an oscillating heat input to the sample. The heat Eows through the sample and out to the heat sink (bath) via a thermal link Kb . The heat-balance equations for the system are as follows (T = dT=dt): Ch Th = p0 + p sin !t − Kh (Th − Ts ) ;
(2.11a)
Cs Ts = Kh (Th − Ts ) − Kb (Ts − Tb ) − Kt (Ts − Tt ) ;
(2.11b)
Ct Tt = Kt (Ts − Tt ) :
(2.11c)
The temperature oscillations are suIciently small to consider the parameters C and K constant. The steady-state solution for these equations consists of two terms, a constant term depending upon Kb and an oscillatory term inversely proportional to the heat capacity of the calorimetric cell: Tt = Tb + p0 =Kb + (pB=!C) sin(!t − ) ;
(2.12)
where C = Cs + Ch + Ct , B is a complicated expression involving quantities from Eqs. (2:11), ◦ and is a phase shift close to 90 under conditions discussed below.
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Fig. 2.3. Frequency dependence of !0 , schematically. Usually, the modulation frequency is chosen in a range where this quantity remains constant. Fig. 2.4. Presentation and equivalent electrical circuit of calorimetric cell containing a substrate, a sample, a heater, and a thermometer (Velichkov, 1992). R = 1=K.
If (i) the heat capacities of the heater and of the thermometer are much smaller than that of the sample, (ii) the sample, the heater, and the thermometer come to equilibrium in a time much shorter than the modulation period, and (iii) the modulation period is much shorter than the sample-to-bath relaxation time s , then, to Frst order in 1=!2 2s and !2 (2h + 2t ), B = [1 + 1=!2 2s + !2 (2h + 2t )]−1=2 ;
(2.13a)
cot = 1=!s − !(h + t ) ;
(2.13b)
where the relaxation times are deFned as s = C=Kb ; h = Ch =Kh ; and t = Ct =Kt . To take into account the Fnite thermal conductivity of the sample, Sullivan and Seidel (1968) considered a sample in the form of a slab heated uniformly on one side by a sine heat Eux. The other side of the slab is coupled to the bath through the thermal conductance Kb . The amplitude of the temperature oscillations in the sample is 0 = p=!C(1 + 1=!2 2s + !2 2 + 2Kb =3Ks )1=2 ;
(2.14)
where various time constants are lumped into ; 2 = 2h + 2t + 2i ; i is the internal relaxation time of the sample depending on its thermal di0usivity and thickness, and Ks is the thermal conductance of the sample in the direction of heat Eow. As a rule, s is two to three orders of magnitude larger than , and the condition !2 2s 1!2 2 is achievable by a proper choice of the modulation frequency. The term 2Kb =3Ks is small owing to the small thickness of the sample. Modulation measurements are usually performed under conditions where the above assumptions are well satisFed, so that the expression for calculating the heat capacity retains the simple form (2.4). When the conditions of the measurements are adequate to the theoretical model, the quantity !0 should not depend on the modulation frequency. To check this prediction, one plots a corresponding graph (Fig. 2.3). The decrease of the quantity !0 at low frequencies shows that the criterion of adiabaticity is not satisFed, while the decrease at high frequencies is due to the inertia of the temperature sensor, a thermocouple or a resistance thermometer. One has to choose the modulation frequency inside the interval where the quantity !0 is constant.
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Later, Velichkov (1992) considered a calorimetric cell consisting of a substrate carrying a sample, a heater, and a thermometer (Fig. 2.4). Such a structure was employed in studies of thin layers deposited directly onto a substrate. The author obtained mathematical solutions concerning the stationary amplitude and phase of the temperature oscillations measured by the thermometer. He presented the results as !0 and tan versus the modulation frequency. The author also stressed the possibility to perform the measurements in a nonadiabatic regime when ! ¡ 10. Earlier, Varchenko and Kraftmakher (1973) examined this regime. In addition, the question arises of how to evaluate contributions from leads and suspensions unavoidable in calorimetric systems. Greene et al. (1972) considered this question. The heat capacity of a wire, Cw , makes a contribution to the total heat capacity given by TC = Cw F(X ), where F(X ) = (sinh 2X − sin 2X )=2X (cosh 2X − cos 2X ) :
(2.15)
Here X = (!Cw =2Kw )1=2 , and Kw is the thermal conductance of the wire. The function F(X ) has the limiting forms F(X ) ≈ 1=3 for X 6 0:5 and F(X ) ≈ 1=2X for X ¿ 2. Thus, as the modulation frequency increases, a smaller fraction of the wire heat capacity contributes to the total heat capacity. This means that heat does not have time to Eow as far down the wire during the shorter period. The quantity F(X ) is both frequency and temperature dependent. Suzuki et al. (1982) found the heat capacity of the addenda of their calorimetric cell to decrease with increasing frequency. This was explained by contributions from leads and suspensions. These contributions were expected to vary as !−1=2 (the case X ¿ 2). To check this conjecture, the authors plotted the measured heat capacity versus !−1=2 . A linear relation was obtained for modulation frequencies above 200 Hz. The slope of the line appeared in good agreement with calculations based on thermal di0usivities and cross-sectional areas of the wires employed. In modulation measurements of other thermophysical properties, the criterion of adiabaticity is of no importance, and the choice of the modulation frequency depends on other considerations. In any case, the temperature sensor must have a low thermal inertia allowing correct measurements of the temperature oscillations.
3. Modulation of heating power The choice of a method to periodically heat the sample depends on its shape and electrical conductivity and on the temperature range of the measurements. At high temperatures, direct electrical heating or electron bombardment is preferable. In studies of nonconducting samples, separate heaters or modulated-light heating are employed. Various methods of heating provide di0erent accuracy of data. For example, the modulated-light heating is usable where there is no need to accurately determine absolute values of the speciFc heat. No data on heating-power oscillations are necessary in modulation measurements of thermal expansivity, temperature derivative of resistance, and thermopower.
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3.1. Direct electric heating As a rule, conducting samples are heated by passing an electric current through them. Several methods are known to modulate the heating power. (1) Heating by an AC current. The modulation frequency is twice the frequency of the current. The amplitude of the power oscillations is equal to the e0ective power. When the current is the only means to heat the sample in a wide temperature range, the amplitude of the temperature oscillations strongly depends on the temperature. The third-harmonic technique is applicable to measure the temperature oscillations. (2) Heating by a DC current with a small AC component superimposed. The modulation frequency equals that of the AC component. The mean temperature and the amplitude of the temperature oscillations in the sample are controlled independently. An important advantage is that the equivalent-impedance method is applicable permitting the speciFc heat to be calculated from parameters of a bridge or potentiometer circuit that are independent of the amplitude of the AC component. (3) Heating by a high-frequency current modulated by the necessary low frequency. Such heating is useful when the sample’s temperature and its oscillations are measured by a thermocouple connected electrically to the sample. In this case, one easily separates the low-frequency signal related to the temperature oscillations from a high-frequency voltage that may appear in the thermocouple circuit. This method is useful for observing relaxation phenomena in the speciFc heat, when temperature oscillations of two frequencies simultaneously occur in the sample. (4) It is also possible to employ rectangular unipolar pulses for heating the samples. When the modulation period is suIciently short, the temperature oscillations are of a triangular shape and their period equals that of the pulses. Pulses of speciFc shape allow one to avoid such a coincidence (Jin et al., 1984). In all the cases, it is easy to precisely determine the AC component of the power applied to the sample. 3.2. Modulated-light heating This elegant method, invented to measure the speciFc heat of nickel near the Curie point (Handler et al., 1967), became very popular. A sample in the form of a foil is placed in a furnace that controls the mean temperature (Fig. 3.1). A light passed through a chopper supplies the AC heating power. A thermocouple and a lock-in ampliFer measure temperature oscillations created in the sample, of about 10 mK. A photocell provides the reference voltage for the ampliFer. The output DC signal of the lock-in ampliFer is proportional to the input AC voltage from the thermocouple. Hence, it is inversely proportional to the heat capacity of the sample. This signal is fed to the Y-input of a plotter. A second thermocouple, with the hot junction inside the furnace, drives the X-input. During the measurements, the temperature of the furnace gradually changes. The modulation frequency that ensures an adiabatic regime of the measurements depends on the thickness of the sample and its thermal di0usivity. Connelly et al. (1971) described this technique in more detail. A nickel single crystal of 3 × 3 × 0:1 mm3 size was placed into a copper block inside a furnace. Helium gas at about
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Fig. 3.1. Modulated-light heating (Handler et al., 1967).
50 kPa ensured the necessary heat exchange. An incandescent lamp powered by a stabilized source provided a radiation constant within 0.4% for several hours. Control of the radiation by a thermopile permitted introducing corrections if necessary. A strip thermocouple of cross-section 70 × 5 m measured the temperature oscillations in the sample. The increase in the mean temperature of the sample above the furnace amounted to 0:5 K. In the frequency range 25–60 Hz, the quantity !0 remained constant within 1%. The rate of heating the furnace was adjustable in the range 0.06 –0:4 K min−1 . In most measurements, the heating-power oscillations were not measured but precautions were taken to make them independent of the sample’s temperature. For this purpose, the samples were covered by a thin layer of graphite or PbS whose spectral absorptance is high and does not depend on temperature. To evaluate the oscillations in the heating power caused by modulated light, Salamon (1970) employed the relation between the mean power supplied to the sample and the temperature di0erence between the sample and the bath: P = KTT , where K is the thermal conductance between them. Ikeda and Ishikawa (1979) also used this approach. They considered the thermal conductance to contain two parts, the conductance of the gas surrounding the sample and that due to the heat Eow through other paths such as thermocouples. The modulated-light heating allows one to build a fully automated measuring setup. For example, Stokka and Fossheim (1982a) have designed such a calorimeter for the range 2–380 K, with a possibility to apply to the samples uniaxial pressures up to 100 MPa. In essence, the setup is very simple (Fig. 3.2). A thermocouple and a lock-in ampliFer measure the temperature oscillations in the sample caused by the absorbed light. The second thermocouple measures the temperature di0erence between the sample and a copper block. A platinum or germanium thermometer and an automatic AC bridge serve to measure the temperature of the block. A microcomputer provides a Eexible program of the measurements. A scanning voltmeter measures the signals corresponding to the mean temperature and the temperature oscillations. All the data are registered by a tape recorder and then processed by a larger computer. Hatta and Minakov (1999) calculated the thickness of a sample necessary to measure its heat capacity within 1% accuracy. For the required accuracy, the sample’s thickness should not exceed (0:3D=!)1=2 , where D is thermal di0usivity of the sample, and ! is the modulation frequency.
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Fig. 3.2. Block diagram of an automated calorimeter employing modulated-light heating (Stokka and Fossheim, 1982a).
Fig. 3.3. Simultaneous measurement of speciFc heat and thermal di0usivity (Salamon et al., 1974).
Salamon et al. (1974) employed modulated-light heating for simultaneous measurements of speciFc heat and thermal di0usivity (Fig. 3.3). Modulated light falls onto a thin sample, while a narrow mask closes a portion of the sample. Under the mask, the temperature oscillations depend on the thermal di0usivity of the sample. Far away from the mask and at proper modulation frequencies, they depend only on the heat capacity of the sample. The frequency of the temperature oscillations is of the order of 10 Hz, and their amplitudes are nearly 1 mK. The measurements have been performed on CoO, SrTiO3 , Cr, and EuO in the range 60 –325 K. Hatta et al. (1985) improved this technique. In the setup, a wide mask gradually shifts the closed portion of the sample by means of a Fne screw.
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Fig. 3.4. Induction heating (Filippov and Makarenko, 1968).
3.3. Other methods 3.3.1. Induction heating Filippov and Makarenko (1968) introduced this method. The sample is placed in an induction furnace fed by a high-frequency current modulated by a low frequency (Fig. 3.4). A micropyrometer measures the mean temperature of the sample, while a photomultiplier detects the temperature oscillations. A blackbody cavity in the sample allows one to avoid corrections for the spectral emittance of the sample. A separate coil serves to evaluate the power dissipated in the sample. The electrical conductivity of the sample, which also must be known, is measured by the four-probe method. Using induction heating, it is diIcult to accurately measure the oscillations of the heating power. Induction heating was employed in noncontact calorimetry (Fecht and Johnson, 1991; Fecht and Wunderlich, 1994; Wunderlich and Fecht, 1993, 1996; Wunderlich et al., 1993), including modulation measurements in space (Wunderlich et al., 1997; Egry, 2000). 3.3.2. Electron bombardment An advantage of electron-bombardment heating is the possibility of using samples of irregular shape. At the same time, it is easy to determine the AC component of the heating power. The power supplied to the sample is modulated as follows. (1) In a saturation regime, the accelerating voltage is modulated, while the electron-beam current remains constant. (2) The temperature of the cathode is modulated or a control electrode is used to modulate the electron current, while the accelerating voltage is constant. (3) The accelerating voltage is periodically switched on and o0, so that the heating-power pulses are rectangular. A drawback of this method is that the power oscillations increase along with the mean heating power. The power dissipated in the sample equals IU , where I is the electron-beam current, and U is the accelerating voltage. A thermocouple can be either welded to the sample or placed into a thin-wall insulating capillary passing through the sample. Welding ensures better thermal coupling but a part of the anode current may appear in the thermocouple circuit. The second method requires longer modulation periods.
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Fig. 3.5. Electron-bombardment heating (Varchenko et al., 1978). A radiative heater controls the mean temperature of the sample.
An additional radiative heater was employed to independently control the mean temperature of the sample and the amplitude of the temperature oscillations (Varchenko et al., 1978). The electron beam was modulated either by a relay powered by a generator of infralow frequency (Fig. 3.5) or by a control electrode between the cathode and the sample. A plotter recorded the temperature oscillations. Using this technique, the speciFc heat of iron was measured in the range 600 –1250 K. 3.3.3. Separate resistive heaters Separate heaters for periodic heating are used mainly at low and middle temperatures and in studies of nonconducting samples. The main requirements for the heater are its small heat capacity and good thermal coupling to the sample. Therefore, microresistors or thin deposited Flms often serve as the heaters. Using AC current, the power oscillations are accurately measured. 3.3.4. Peltier heating This method has an advantage that no DC temperature increment is introduced. McNeill (1962) employed AC Peltier heating in measurements of the thermal di0usivity of thermoelectric materials. Johansen (1987) used a Peltier element to periodically heat the sample in a modulation dilatometer. Moon et al. (2000) have described a Peltier microcalorimeter.
4. Measurement of temperature oscillations Various methods are applicable to detect temperature oscillations. First, they were determined from oscillations in the resistance of the sample, thermionic current, or radiation from it. Later, separate temperature sensors, thermocouples and resistance thermometers, served for this purpose. Oscillations of the thermionic current are usable only in special cases, and this technique is now out of use. A choice of an adequate method to measure the temperature oscillations is very important. It depends on the temperature range, shape and electrical conductivity of the sample, and the modulation frequency.
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Fig. 4.1. Use of supplementary current of a frequency much higher than that of temperature oscillations (Gerlich et al., 1965).
4.1. Use of oscillations in the sample’s resistance An advantage of this method is that the measurements are possible at relatively low temperatures where the radiation from the sample is negligible. In addition, high modulation frequencies are applicable because one avoids the problem of the thermal inertia of a temperature sensor. The only drawback of the method is the necessity to know the temperature dependence of the sample’s resistance and of its temperature derivative. The method is inapplicable when these quantities exhibit anomalies. To measure temperature oscillations, three techniques are known: the supplementary-current method, the third-harmonic technique, and the equivalent-impedance method. Evidently, in all the cases the temperature derivative of the sample’s resistance, R , enters the expressions for the determination of the temperature oscillations. 4.1.1. Supplementary-current method Corbino (1910) invented this method long ago. He observed a DC voltage drop across a sample heated by an AC current when a supplementary AC current of a frequency equal to that of the temperature oscillations passed through it. The oscillations in the sample’s resistance are R 0 sin 2!t. A supplementary current i sin(2!t − ) causes a DC voltage drop across the sample that equals (iR 0 =2) cos . It reaches a maximum when the supplementary current has the same phase as the temperature oscillations. The supplementary current is much smaller than the heating current. Gerlich et al. (1965) employed a supplementary current of frequency 320 Hz, much higher than that of the temperature oscillations (Fig. 4.1). A 1-Hz current passes through the sample, setting up temperature oscillations at 2 Hz. A potentiometer balances the main component of the 320-Hz voltage across the sample. A lock-in ampliFer measures the component related to the temperature oscillations, with the frequency 320 Hz as a reference. A 2-Hz signal obtained from the lock-in ampliFer proceeds, through a Flter, to the Y-input of an oscilloscope. The X-input of the oscilloscope is connected to an oscillator of frequency 1 Hz. The Lissajous pattern is photographed and the speciFc heat is evaluated from its shape and the temperature derivative of the sample’s resistance.
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Fig. 4.2. Measurement of third-harmonic signal (Filippov, 1960; Rosenthal, 1961).
A modiFcation of the method consists in using a supplementary current of a frequency close to that of the temperature oscillations (Kraftmakher and Tonaevskii, 1972). This results in a di0erence-frequency voltage across the sample, that is proportional to the oscillations in the resistance and to the supplementary current. An advantage of the method is that the di0erence-frequency signal depends only on the oscillations of the resistance. The method served to directly determine the temperature coeIcient of the speciFc heat. In this case, it was necessary to measure a weak second-harmonic component of the temperature oscillations in the presence of a much stronger fundamental signal. 4.1.2. Third-harmonic technique Corbino (1911) found that when an AC voltage is applied to a sample, the current through it contains a third-harmonic component proportional to the temperature oscillations and the temperature derivative of the sample’s resistance. He measured this component by compensating the fundamental-frequency current and observing the Lissajous pattern by means of a cathode-ray tube. When an AC voltage U sin !t is applied to a sample, the temperature oscillations in it, under adiabatic conditions, are = −0 sin 2!t. The current through the sample equals i = (U sin !t)=(R0 − R 0 sin 2!t). The oscillations of the sample’s resistance are small, so that the current obeys the relation i = U (sin !t + 0 sin !t sin 2!t)=R0 = U [sin !t + ( 0 =2) cos !t − ( 0 =2) cos 3!t]=R0 ;
(4.1)
R =R0 .
where = If the sample is fed through a suIciently high resistance, the amplitude of the third-harmonic voltage across the sample, under adiabatic conditions, is V3 = I 3 R0 R =8mc! = U 3 R =8R20 mc! ;
(4.2)
where I and U are the amplitudes of the fundamental-frequency current through the sample and of the voltage across it. Filippov (1960) measured the third-harmonic signal by a bridge circuit with the sample as one of its arms (Fig. 4.2). The bridge is balanced at the fundamental frequency, and the third-harmonic output signal is observed by means of a selective ampliFer (not shown in the
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Fgure) and an oscilloscope. A variable capacitor shunting one of the arms serves to compensate the additional fundamental-frequency voltage caused by the temperature oscillations. Rosenthal (1961, 1965) studied the temperature oscillations in thin wires over a wide frequency range and presented the data as polar diagrams 0 (). Holland (1963) employed the third-harmonic method to measure the speciFc heat of titanium over the range 600 –1345 K. Titanium undergoes a structural transition at 1155 K, from a hexagonal close-packed low-temperature form to a body-centered cubic high-temperature form. The sample was a 0.25-mm Flament, 40 cm long. Potential probes, 25-m tungsten wires, were placed 10 cm from each end of the Flament. The mean temperature of the sample was computed from the electric power dissipated in the sample. In the temperature range of the measurements, the total emissivity of titanium is nearly constant. A Kelvin bridge was used to balance the fundamental voltage, together with any third harmonic originating in the source of the heating current. An electrodynamometer served as the lock-in detector. A frequency tripler provided the signal for its Fxed coil. The mean torque on the moving coil is proportional to the averaged product of the currents in the two coils. When the current in the Fxed coil is sinusoidal, only a current of the same frequency in the moving coil produces a torque with a nonzero time average. Smith (1966) used this technique to measure the speciFc heat of a germanium whisker in ◦ the range 350 –650 C. Skelskey and Van den Sype (1970) employed the third-harmonic method in measurements of the speciFc heat of gold. Birge and Nagel (1985, 1987) and Birge (1986) used this method in measurements of frequency-dependent speciFc heat of supercooled liquids over the range 0:01 Hz–6 kHz. Jung et al. (1992) described a fully automated calorimeter employing the third-harmonic technique up to 10 kHz. Measurements with di0erent modulation frequencies are aimed at a search for relaxation phenomena in speciFc heat. The method, often referred to as the 3! technique, was also used in many other studies (Dixon and Nagel, 1988; Dixon, 1990; Jeong et al., 1991; Jeong and Moon, 1995; Menon, 1996; Ema and Yao, 1997). 4.1.3. Equivalent-impedance method Radio engineers found long ago that a temperature-sensitive resistor through which a DC current is Eowing displays an equivalent impedance depending, in particular, on its heat capacity (Griesheimer, 1947; Jones, 1953; Van der Ziel, 1958). However, a time elapsed before thermophysicists realized the advantages of speciFc-heat measurements based on this principle. This technique is convenient for measurements on wire samples at high temperatures (Kraftmakher, 1962, 1994c). When a current I = I0 + i sin !t (iI0 ) heats a wire sample, its resistance follows the relation R = R0 + R = R 0 sin(!t − ’) = R0 + R 0 cos ’ sin !t − R 0 sin ’ cos !t :
(4.3)
The voltage drop across the sample is IR = (I0 + i sin !t) × (R0 + R 0 cos ’ sin !t − R 0 sin ’ cos !t) = I0 R0 + I0 R 0 cos ’ sin !t − I0 R 0 sin ’ cos !t + iR0 sin !t +iR 0 cos ’ sin2 !t − iR 0 sin ’ sin !t cos !t :
(4.4)
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Fig. 4.3. Real and imaginary parts of equivalent impedance caused by temperature oscillations. !=!0 = B=A = tan ’, ◦ where !0 is a frequency for which ’ = 45 . As a rule, measurements are performed under conditions where 2 ABR0 , and B = !R C. Fig. 4.4. Bridge circuit for measuring speciFc heat of wire samples (Kraftmakher, 1962, 1994c). Because of temperature oscillations in the sample, an adjustable capacitor is necessary to balance the bridge.
Excluding the DC terms and neglecting the small AC terms, the AC voltage drop across the sample is V = iR0 sin !t + I0 R 0 cos ’ sin !t − I0 R 0 sin ’ cos !t :
(4.5)
This voltage contains two components, one in phase with the AC component of the current, and the other quadrature lagging. The impedance of the sample Z describes the amplitude and phase relations between the AC components of the current through and the voltage drop across the sample. It may be written as a complex quantity Z = R0 + A − iB. The quantities A and B are obtainable from (4.5) and (2.9b) and by dividing the AC voltage across the sample by the AC component of the current: Z = R0 + (2I02 R0 R =mc!) sin ’ cos ’ − i(2I02 R0 R =mc!) sin2 ’ :
(4.6)
The ratio B=A equals tan ’, so that at high frequencies the additional real part of the equivalent impedance, A, is much smaller than the imaginary part, B (Fig. 4.3). An equivalent electric circuit may be represented by a resistor and a capacitor connected in series: Z = R∗ − i=!C ∗ . The resistance R∗ equals R0 + A, and the capacitance C ∗ = mc=2I02 R0 R . In the adiabatic regime, AR0 , so that Z = R0 − i(2I02 R0 R =mc!). The equivalent capacitance C ∗ does not depend on the frequency of the temperature oscillations. However, the large equivalent capacitance makes this approach impractical. To avoid this diIculty, the equivalent circuit can be represented as a resistor R and a capacitor C connected in parallel: Z1 = R=(1 + !2 R2 C 2 ) − i!R2 C=(1 + !2 R2 C 2 ) :
(4.7)
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Fig. 4.5. Potentiometer circuit that eliminates cold-end e0ects (Kraftmakher, 1966b). Only the central portion of the sample is involved in the measurements.
∼ 1. Hence, Under suIciently high modulation frequencies, !2 R2 C 2 1; AR0 , and sin2 ’ = 2 R = R0 ; B = !R C, and the main relation for the equivalent-impedance technique is
mc = 2I02 R =!2 RC :
(4.8)
When deriving the above expressions, the temperature derivative of the sample’s resistance was assumed to be positive. The heat capacity of the sample is thus available from the parameters of the equivalent impedance, R and C. A bridge circuit, one arm of which is shunted by a variable capacitor, serves for the measurements (Fig. 4.4). A selective ampliFer tuned to the modulation frequency ensures high sensitivity even when the temperature oscillations in the sample are small. For very small temperature oscillations, it is necessary to employ a lock-in ampliFer. Using a dual-channel lock-in ampliFer, one of the detectors is set to be sensitive to the in-phase component of the input voltage and the other to the quadrature component. Signs of the output DC voltages of the detectors show whether an increase or decrease of the resistor R and capacitor C are needed to complete the compensation. This method allowed us to determine the speciFc heat of tungsten at temperatures up to 3600 K (Kraftmakher and Strelkov, 1962). Kraev (1967) found that the above bridge circuit is suitable for measuring speciFc heat even when only an AC current passes through the sample. He measured the speciFc heat of tungsten up to 3000 K. A drawback of this approach is a strong temperature dependence of the amplitude of the temperature oscillations. The bridge circuit is inapplicable in measurements at relatively low temperatures when the cold-end e0ects become signiFcant. In this case, one has to use very long wire samples or to heat the current leads to which the sample is welded. As a good alternative, a potentiometer circuit is usable to measure the heat capacity of the central portion of the sample (Fig. 4.5). A DC current with a small AC component heats the wire sample. The potential probes are much thinner than the sample and cause no signiFcant temperature changes at the points where they are welded to the sample. To satisfy this requirement more reliably, one can heat the probes by passing an additional current through them. The voltage drops across the resistors R2 and R3
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are compared by means of a selective ampliFer. The resistor R1 and capacitor C1 are adjusted to obtain a proper amplitude of the AC current in the compensation circuit and a phase strictly coinciding with that of the AC component of the heating current. The ampliFer is then switched to measure the equivalent impedance of the central portion of the sample. As in the case of the bridge circuit, the parameters of the equivalent impedance, R and C, do not depend on the AC component of the heating current. An oscilloscope indicates the balance, and a lock-in ampliFer is applicable. The resistance R1 is much larger than R, so that the amplitude and the phase of the current in the compensation circuit are independent of R and C. Expression (4.8) is valid but all the quantities correspond to the central portion of the sample. Using the equivalent-impedance method, the quantities m; I0 ; !; R, and C are determined very accurately. The mass of the sample is measured with an error smaller than 1%, whereas errors in the other quantities are of the order of 0.1%. The total accuracy of the speciFc heat depends mainly on the accepted values of R . Most favorable is the case where the temperature dependence of the sample’s resistance is nearly linear, so that the temperature derivative R weakly depends on temperature. For example, for tungsten and molybdenum the temperature dependence of this quantity is about 1% per 100 K. On the other hand, the method is inapplicable in studies of phase transitions accompanied by anomalies in electrical resistivity. 4.2. Photoelectric detectors Many workers determined the temperature oscillations from the sample’s radiation. Lowenthal (1963) measured the speciFc heat of tungsten, tantalum, molybdenum, and niobium in the 1200 – 2400 K range. An AC current heated the samples, and a photomultiplier served to measure the temperature oscillations. The dependence of the photomultiplier’s current on the temperature of the sample was assumed to have the form I = AT n , with n weakly dependent on temperature. Hence, the amplitude of the temperature oscillations obeys the relation 0 = TV=nV0 ;
(4.9)
where V0 and V are the DC and AC components of the photomultiplier’s output voltage. Filippov et al. (1964) and Filippov and Yurchak (1965) considered the dependence of the photomultiplier’s current on the temperature of the sample to be I = B exp(−A=T ). In this case, 0 = T 2 V=AV0 :
(4.10)
However, neither of these methods suIces because the quantities A; B, and n are temperature dependent (A and B depend on the e0ective wavelength and on the spectral emittance of the sample). One can avoid these diIculties and gain an important additional advantage by employing samples with blackbody cavities. For all such samples maintained at equal mean temperatures, the relation between oscillations of the radiation from the cavities and temperature oscillations is the same. This enables one to directly compare temperature oscillations in various samples, without calibration of the photosensor. With a sample of known speciFc heat, comparative measurements of speciFc heat are possible. Tungsten and platinum may serve as reference materials for such measurements. Their speciFc heat was determined using all the calorimetric methods already known. Tungsten is a good reference material for temperatures up
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Fig. 4.6. Use of samples with blackbody cavities (Kraftmakher, 1992a, b). Under compensation, mc = KI0 RC, where K depends only on the sample’s temperature.
to 3000 K and platinum up to 1500 K. The high-temperature speciFc heat of other metals and alloys is not so well known, and comparative measurements of their speciFc heats are useful. Akhmatova (1965, 1967) employed comparative measurements in studies of molten metals. The samples were placed in a niobium capillary heated by an AC current. The oscillations of the radiation from the empty and Flled capillary kept at the same mean temperature were compared. This method provides the ratio of the heat capacities of the molten metal and the capillary. A compensation circuit, whose balance is independent of the AC component of the heating power, may enhance the accuracy of such measurements. In the compensation circuit, a sample with a blackbody cavity is heated by a DC current I0 with a small AC component (Fig. 4.6). To exclude cold-end e0ects, only the central portion of the sample restricted by thin potential probes serves for the measurements. The mean temperature and the amplitude of the temperature oscillations are constant throughout the central portion. In the adiabatic regime, the speciFc heat is given by mc = 2I0 U=!0 ;
(4.11)
where m and U are the mass of the central portion of the sample and the AC voltage across it. The blackbody cavity is projected onto a photodiode. The AC voltage across the load resistor of the photodiode is proportional to the temperature oscillations in the sample: V1 = K1 0 , where K1 is a proportionality factor. Owing to the blackbody cavity, this factor depends only upon the mean temperature of the sample. The potential probes are connected to the input of an integrating RC circuit. The output voltage of this circuit is ampliFed by a wide-band ampliFer and fed to the load resistor of the photodiode. The AC voltage at the capacitor C equals U=!RC (!2 R2 C 2 1). The compensation voltage is V2 = K2 U=!RC, with K2 a proportionality factor. Under compensation, V1 = V2 , i.e., mc = 2K1 I0 RC=K2 = KI0 RC :
(4.12)
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The coeIcient K =2K1 =K2 is the same for all the samples of a given mean temperature. Only the variable capacitor C provides the compensation. Therefore, for two samples one obtains m1 c1 =m2 c2 = I01 C1 =I02 C2 ;
(4.13)
where I01 and I02 are DC currents providing equal mean temperatures of the samples, and C1 and C2 are the values of the capacitance corresponding to the compensation. Hence, comparative measurements are possible on any conducting samples provided with a blackbody cavity. In addition, the total radiation from the cavity is used instead of radiation in a narrow spectral band. This compensates the decrease in the radiant Eux due to the small size of the cavity. The radiation from the cavity and from a standard strip lamp is in turn projected, by the same optical system, onto a photodiode or another photosensor. The standard lamp serves to determine the mean temperature of the samples. Both samples and the standard lamp are located in a vacuum chamber and are replaced by means of a turn-plate. No corrections are thus needed for the reEectance and absorption of the radiation in the chamber’s window. 4.3. Thermocouples and resistance thermometers At middle and low temperatures, thermocouples and resistance thermometers are the most reliable tools for measuring the mean temperature of the sample and the temperature oscillations in it. Thermocouples are widely used with modulated-light heating or separate electrical heaters. In some cases, the thermocouples were formed by deposited thin Flms, 10 –100 nm thick. A good thermal coupling and low thermal inertia are thus achievable allowing measurements over wide frequency ranges. With thermocouples, it is possible to measure temperature oscillations in the range 1–10 mK, the smallest value was 0:1 mK (Bonilla and Garland, 1974). Temperature oscillations of the order of 1 K are measurable at liquid helium temperatures by means of resistance thermometers (Mehta and Gasparini, 1997, 1998; Mehta et al., 1999). When using thermocouples, one has to ensure good thermal coupling of the junction to the sample and to make the additional heat capacity be negligible. Craven et al. (1974) measured the speciFc heat of TTF-TCNQ on thin platelets, typically 0:3 × 0:02 × 10 mm3 . The samples were mounted on a pair of thermocouples formed by spot-welded 25-m chromel and alumel wires, Eattened to 5 m in the junction region. Pulses of light from a quartz-iodine lamp heated the sample. One of the junctions of the crossed thermocouples detected the temperature oscillations of the system, while the other formed the cold junction of a thermocouple with its hot junction in an ice bath. GarFeld and Patel (1998) described a method of preparing thin thermocouples. Ultrasmall thermocouple probes were used for a scanning thermal proFler (Williams and Wickramasinghe, 1986). Usually, a resistance thermometer is included in a DC bridge or potentiometer circuit. Since the resistance of the thermometer oscillates according to the temperature oscillations, the AC output voltage of the bridge immediately provides the necessary data. As a rule, a lock-in ampliFer measures this voltage. To calibrate a resistance thermometer, one needs a precisely calibrated thermometer, usually commercially available. Metallic thermometers, such as a platinum thermometer, satisfy the relation T = An Rn . However, they are impracticable at temperatures below 10 K because of their low sensitivity. This is caused by the residual resistance due to dissolved impurities and defects
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Fig. 4.7. Principle of lock-in detection. Averaged output voltage is proportional to the AC input signal of a frequency coinciding with that of the reference and to the cosine of the phase shift between them.
of the crystal structure. At low temperatures, semiconductors provide much higher sensitivity that increases with decreasing temperature. For such thermometers, a more suitable relation is 1=T = An (ln R)n = A0 + A1 ln R + A2 (ln R)2 + · · · : Semiconducting thermometers, thermistors or usual carbon resistors, are applicable also at room temperatures. To check the possible inEuence of self-heating, the calibration is carried out with di0erent measuring currents. The calibration should take into account the temperature-dependent sensitivity of the thermometer, the measuring current, and the ampliFer’s gain. 4.4. Lock-in detection of periodic signals Lock-in detection is an excellent tool for measuring weak periodic signals in the presence of signals of other frequencies or noise. The method rests on the exact knowledge of the frequency of the expected signal. This periodic signal is measured by a detector controlled by a reference voltage taken from the oscillator governing the process under study. The frequency of the expected signal therefore strictly coincides with that of the reference, while the phase shift between them remains constant. In the modulation techniques, the reference voltage is supplied either by the source of the modulated power or by a special sensor, e.g., a photocell when using modulated-light heating. The output signal of the detector is averaged over a time suIciently long to suppress signals of other frequencies and noise. The e0ective bandwidth of a lock-in detector is inversely proportional to the averaging time. The detector is thus always tuned to the expected signal and has a readily adjustable bandwidth. The operation of a lock-in detector may be explained as follows. An electronic switch controlled by the reference voltage periodically changes the polarity of the signal fed to an integrating RC circuit (Fig. 4.7). A DC output voltage at the output of the RC circuit appears only when the signal contains a component of the reference frequency. The DC voltage is proportional to this component and to the cosine of the phase shift between it and the reference. The detector incorporates an adjustable phase shifter to achieve the maximum output voltage. When the signal contains no component of the reference frequency, the averaged output voltage remains zero. The reason for this is the absence of Fxed phase relations between the reference and signals of other frequencies or noise. AmpliFers employing lock-in detection are called lock-in ampliFers. Owing to the narrow bandwidth, they provide high sensitivity and noise immunity. Some types of lock-in ampliFers ◦ incorporate two detectors governed by references with phases shifted by 90 . This makes it
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possible to measure the signal and its phase shift relative to the reference. Dual-channel lock-in ampliFers are necessary when the signal to be measured or balanced contains both in-phase and quadrature components, e.g., in the equivalent-impedance technique. A lock-in detector is eIcient for measuring small phase changes. In this case, the phase shift ◦ between the signal and the reference is set at 90 . Under such conditions, the output voltage of the detector is zero, but the sensitivity to phase changes is the best. 5. Modulation calorimetry 5.1. Methods of calorimetry, a brief review Calorimetric measurements seem, at Frst glance, to be simple and straightforward. By deFnition, one has to supply some heat to the sample and to measure the corresponding increment in its temperature. However, no simple solution for this problem exists in a wide temperature range. First, the accuracy of temperature measurements in various temperature ranges is very di0erent. Second, it is impossible to completely avoid uncontrollable heat exchange between the sample and its surroundings when the sample’s temperature is far from room temperature. During many years of development, the following methods were proposed to solve the problem. (1) All possible precautions are undertaken to reduce the unwanted heat exchange between the sample and its surroundings (adiabatic calorimetry). (2) The enthalpy of the sample is measured instead of the speciFc heat: the sample heated up to a high temperature drops into a calorimeter usually kept at room temperature, and the heat released from the sample is measured (the drop method). (3) Shortening the time of the measurements minimizes the inEuence of any uncontrollable heat exchange (modulation, pulse, and dynamic techniques). (4) The heat exchange between the sample and its surroundings is taken into account and involved in the measurements of speciFc heat (the relaxation method). 5.1.1. Adiabatic calorimetry Adiabatic calorimetry reduces to a minimum any heat exchange between the calorimeter and its environment. For this purpose, the calorimeter is surrounded by a shield whose temperature is kept equal to that of the calorimeter during the entire experiment, the so-called adiabatic shield (e.g., Kraftmakher and Strelkov, 1960). The calorimeter and the shield are placed in a vacuum chamber, so that only radiative heat exchange occurs. An electrical heater heats the calorimeter. A resistance thermometer or a thermocouple measures the temperature. The heat supplied to the calorimeter and the temperature increment are thus accurately known. To reduce the heat Eow through the electrical leads, they are thermally anchored to the adiabatic shield. With a temperature di0erence between the calorimeter and the shield TT , the power of the radiative heat exchange is proportional to T 3 TT . Adiabatic calorimetry, being an excellent technique at low and middle temperatures, fails therefore at high temperatures. Continuous operation with enhanced heating rate reduces the role of heat losses. Braun et al. (1968) have developed an adiabatic calorimeter with continuous heating for the range 300 –1900 K. It is capable of measurements of the speciFc heat of solids (with an error of 2%) and liquids (3%), and latent heats of phase transitions in solids (0.5%), and of melting
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Fig. 5.1. SimpliFed diagram of setup for drop calorimetry (Glukhikh et al., 1966).
(1.5%). Three modes of operation are feasible: (i) the applied power is constant, while the heating rate is inversely proportional to the heat capacity of the sample; (ii) the heating rate is kept constant, so that the applied power is proportional to the heat capacity; (iii) with the heater switched o0, the temperature di0erence between the calorimeter and the thermal shield is monitored. Buckingham et al. (1973) designed a high-precision calorimeter with continuous heating. It was designed for measurements near phase transitions, where the rate of change of speciFc heat and of thermal relaxation time may become very large. Kagan (1984) has reviewed adiabatic calorimetry for middle and high temperatures. Stewart (1983), Gmelin (1987, 1999) and Suga (2000) have reviewed calorimetric techniques for low temperatures. 5.1.2. Drop method This technique was developed for measurements at high temperatures. A sample placed in a furnace is heated up to a selected temperature measured by a thermocouple or an optical pyrometer (Fig. 5.1). Then it drops into a calorimeter kept at a temperature convenient for measurements of the heat released from the sample, usually room temperature. A resistance thermometer measures the increment in the temperature of the calorimeter, which is proportional to the enthalpy of the sample. The calorimetric measurements are thus carried out under conditions most favorable for reducing any unwanted heat exchange. The price for this gain is that the result of the measurements is the enthalpy, instead of the speciFc heat. The speciFc heat is obtainable as the temperature derivative of the enthalpy. When the speciFc heat is weakly temperature dependent, the method is quite adequate. The situation becomes more complicated when the speciFc heat varies in narrow temperature intervals but the corresponding changes in the enthalpy are too small to be determined precisely. An additional disadvantage appears when Frst-order phase transitions occur at intermediate temperatures and thermodynamic equilibrium in the sample after cooling is doubtful. Drop calorimetry was proposed when there was no alternative at high temperatures. Nevertheless, it remains useful until today, especially for measurements on nonconducting materials (for details see Ditmars, 1984).
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Fig. 5.2. Pulse calorimetry at high temperatures (Rasor and McClelland, 1960a).
A modiFcation of the drop technique is levitation calorimetry, where the sample is levitated and heated by a high-frequency electromagnetic Feld (for reviews see Chekhovskoi, 1984, 1992). No container is thus needed, and the temperature range of the measurements can be extended above the melting points of refractory materials. For instance, Arpaci and Frohberg (1984) measured the speciFc heat of tungsten up to 4000 K. 5.1.3. Pulse and dynamic techniques The inEuence of an unwanted heat exchange between a sample and its surroundings is proportional to the time of the measurements. A decrease of this time is an eIcient method even at very high temperatures. This approach is applicable whenever it is possible to heat the sample and to measure its temperature rapidly. Conducting samples heated by an electric current or by electron bombardment are well suited for such measurements. The temperature of the sample is measured through its resistance or radiation. When the heat losses cannot be completely avoided, they are taken into account. Pulse calorimetry employs small increments in the sample’s temperature, and the result of one measurement is the speciFc heat at a single temperature. A furnace or electric heating provides the initial temperature of the sample. Rasor and McClelland (1960a) have designed an apparatus with a sample placed in a graphite furnace to achieve the initial temperature (Fig. 5.2). A photomultiplier senses the temperature increment caused by an electric pulse. By means of a four-channel oscilloscope, the heating current I , the voltage drop across the sample U , the temperature of the sample T , and the heating rate Th = (dT=dt)h are recorded simultaneously. With heat losses neglected, the heat capacity of the sample is mc = IU=Th :
(5.1)
Rasor and McClelland (1960b) measured the speciFc heat of molybdenum, tantalum, and graphite at high temperatures, up to 3920 K for graphite. They observed, for the Frst time, a strong nonlinear increase in the speciFc heat of refractory metals. The authors have concluded that it is too high to be attributed to point-defect formation. The dynamic technique consists in heating the sample over a wide temperature interval. The heating power and the sample’s temperature are measured continuously during the run. If the
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29
Fig. 5.3. Dynamic calorimetry developed by Cezairliyan and coworkers. In one run, speciFc heat, electrical resistivity, normal spectral emittance, and hemispherical total emittance are measured in wide temperature interval. This technique was successfully employed in studies of many high-melting-point metals and alloys. The speciFc heat of tungsten was measured up to 3600 K (Cezairliyan and McClure, 1971).
heat losses from the sample are signiFcant, they are measured during the cooling period and are taken into account. These data are suIcient to evaluate the speciFc heat in the whole temperature range. Cezairliyan and coworkers at the National Bureau of Standards have developed a convenient and accurate subsecond technique for temperatures up to 3600 K (for reviews see Cezairliyan, 1984, 1988, 1992). The temperature of the samples with blackbody models was measured by an optical pyrometer (Fig. 5.3). The power-balance equations for the heating and cooling periods are as follows: mcTh + P = IU ;
(5.2a)
mcTc + P = 0 :
(5.2b) Tc
Here P is the power of the heat losses from the sample, and = (dT=dt)c is the cooling rate after ending the heating. From these relations, the heat capacity of the sample is mc = IU=(Th − Tc ) ;
(5.3)
where Th and Tc relate to the same temperature of the sample. Righini et al. (1985, 1993, 1999) also developed a setup for dynamic measurements and measured speciFc heat and other thermophysical properties of metals at high temperatures. A setup described by DobrosavljeviRc and MagliRc (1989) employs wire samples and heating rates up to 1500 K s−1 . The setup consists of a vacuum chamber, an electric power circuit, measuring and control devices, and a computer. The sample, 2 mm in diameter, has a total length of about 200 mm. The central portion of the sample is 20 mm long. Three 0.05 or 0.1-mm thermocouples are spot-welded at the center and symmetrically at 10-mm separations on both sides. The thermocouple legs serve also as potential leads to measure the voltage drop across the central portion of the sample. The data are collected at a 1-kHz sampling rate. Contact temperature measurements in the presence of an electric current pose a problem because it is impossible to position both thermoelectrodes exactly along the same equipotential line.
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Comparison of the last thermocouple reading during the heating period and the Frst reading after ending the current enables one to introduce necessary corrections. In the 300 –1900 K range, the maximum uncertainties were estimated to be 3% in the speciFc heat and 1% in the electrical resistivity. For other results by this technique see MagliRc (1979), DobrosavljeviRc et al. (1989), DobrosavljeviRc and MagliRc (1991), MagliRc and DobrosavljeviRc (1992), MagliRc et al. (1994, 1995=1996, 1997), PeroviRc et al. (1996), VukoviRc et al. (1996), and MiloYseviRc et al. (1999). Cezairliyan and Righini (1996) have reviewed high-speed optical pyrometry. 5.1.4. Relaxation method This technique employs measurements of the cooling (or heating) rate of a sample whose temperature di0ers from that of the surroundings (Bachmann et al., 1972; Denlinger et al., 1994). This rate depends on the temperature di0erence, the heat capacity of the sample, and the heat transfer coeIcient, i.e., the temperature derivative of the heat losses from the sample. One of the methods employs a calorimeter in which a sample under study and a reference one are placed in turn. The temperature dependence of the heat losses from the calorimeter remains the same, and it is easy to calculate the ratio of the speciFc heats of the two samples. In the step method, the calorimeter is Frst brought to a temperature T = T0 + TT , slightly higher than that of the surroundings, T0 . After ending the heating, the temperature of the calorimeter decays exponentially: T = T0 + TT exp(−t=) ;
(5.4)
where = C=K is the relaxation time, C is the sum of the heat capacities of the sample and of the calorimeter itself, and K is the heat transfer coeIcient. The determinations of speciFc heat thus include measurements of the relaxation time and of the heat transfer coeIcient. Under steady-state conditions, P = KTT , where P is the power applied to the calorimeter that is necessary to increase its temperature by TT . The temperature increment is small, so that the linear relation is valid. The internal time constant (inside the sample and between the sample, the heater, and the thermometer) is much shorter than the external time constant . In the sweep method, a wide temperature range is covered in one run. For this purpose, the steady-state temperature of the calorimeter in the temperature range is determined beforehand as a function of the heating power. Then the heat capacity of the sample becomes available from the cooling curve. The relaxation method is applicable even to samples as small as a few micrograms. The relaxation technique was employed by Zinov’ev and Lebedev (1976) in measurements on tungsten in the range 2400 –3600 K. After heating a wire sample to a high temperature, the heating current was switched o0 and the cooling curve was measured by means of a photomultiplier and an oscilloscope. The heat capacity obeys the relation mc = −P=Tc :
(5.5)
Here P is the power of heat losses from the sample, and Tc is the cooling rate at this temperature. The results obtained by the authors coincide fairly well with those from modulation measurements (Kraftmakher and Strelkov, 1962).
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Fig. 5.4. Relaxation technique employing light heating (Hatta, 1979).
Fig. 5.5. Apparatus for rapid-heating experiments (Pottlacher et al., 1991, 1993).
Hatta (1979) designed a relaxation calorimeter employing light heating (Fig. 5.4). Ema et al. (1993) and Yao et al. (1998) described calorimeters operating in both relaxation and modulation modes. 5.1.5. Rapid-heating experiments The rapid-heating technique allows measurements of thermophysical properties to be made over wide temperature intervals, including the liquid state. Many new results were obtained with this method (for reviews see Lebedev and Savvatimskii, 1974; Gathers, 1986; Cezairliyan et al., 1990). Pottlacher et al. (1991, 1993) reported experiments with heating rates greater than 109 K s−1 . Starting at room temperature, the measurements are performed far into the liquid phase of the metal under study, up to 10 000 K. In the apparatus developed, the energy is stored in a 5.4-F capacitor, with the charging voltage 4 –8 kV (Fig. 5.5). The wire samples are typically 40 mm long and 0:25 mm thick. Water serves as the ambient medium to avoid peripheral discharges. The highest pressure in the vessel is 2 × 108 Pa. An initial pulse triggers a Eashlight for background illumination of the sample, while a three-electrode spark gap triggers the main discharge. The quantities measured during the entire run are as follows: (i) the current through the sample, by means of an induction coil; (ii) the voltage drop across the sample, using
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a coaxial voltage divider; (iii) the radiance temperature of the sample, by a fast pyrometer; and (iv) the Fnal volume of the sample, employing a shadowgraph technique with a 30-ns exposure. The melting temperatures serve as the calibration points, and the emissivity of the sample’s surface is considered independent of temperature. Special care should be taken to avoid superheating the samples, which is quite probable in rapid-heating experiments. The measurements provide the enthalpy, the electrical resistivity, and the volume expansion of the sample. The authors pointed out that more accurate results for the solid phase are available from static measurements. However, rapid heating allows one to perform measurements far above the melting point. This technique has the potential to monitor the vacancy equilibration and hence to reveal vacancy contributions to the enthalpy and electrical resistivity of metals. This group has also designed a microsecond-resolution system (Kaschnitz et al., 1992). Heating rates of 107 –108 K s−1 permit more accurate measurements than do faster systems. Wire or tube-shaped samples are resistively heated using an RCL discharge circuit. Energy is stored in a capacitor, 240 –500 F, which may be charged up to 10 kV. Typically, the heating current is about 5000 A and the pulse is 80 s long. To measure the temperature of the sample, a lens produces its magniFed image at the rectangular entrance of an optical Fber. The light passes through the Fber and enters a photodiode detector. The detector is self-calibrated with the plateau of the melting transitions. The thermal expansion of the samples is determined photographically, by means of a Kerr cell. The authors estimated the uncertainty of the data to be 3% for the enthalpy and 3% for the electrical resistivity, without corrections for the thermal expansion of the samples. 5.2. Modulation calorimetry at high temperatures 5.2.1. Wire samples In the Frst modulation measurements of speciFc heat, including those by Corbino (1910, 1911), an AC current passing through wire samples heated them. In this case, the amplitude of the heating-power oscillations equals the mean applied power. To measure the temperature oscillations, Corbino developed two techniques, the supplementary-current method and the third-harmonic method. A good Fnding turned out to be the equivalent-impedance technique. It employs a bridge or potentiometer circuit whose balance is independent of the AC component of the heating current. The heat capacity of the sample is available from the resistance and capacitance corresponding to the balance. A disadvantage of this technique is the necessity to know the sample’s resistance and its temperature derivative. This drawback is present in all methods based on determinations of the temperature oscillations through the sample’s resistance. The bridge circuit was used in measurements of the speciFc heat of tungsten in the range 1500 –3600 K (Kraftmakher and Strelkov, 1962). Tungsten has a high melting point, and for a long period it served for the fabrication of Flaments for incandescent lamps and cathodes for vacuum tubes. The temperature dependence of its resistivity was carefully studied and now is well known. The measurements provided a good opportunity to check the equivalent-impedance technique. In the range 1500 – 2500 K, the results obtained were in good agreement with existing data. At higher temperatures, a strong nonlinear increase in the speciFc heat was observed. This strong nonlinear increase was conFrmed by various calorimetric techniques (Fig. 5.6). The equivalent-impedance method
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33
Fig. 5.6. SpeciFc heat of tungsten at high temperatures. 1—curve representing data from modulation method (Kraftmakher and Strelkov, 1962), pulse calorimetry (A0ortit and Lallement, 1968), and relaxation measurements (Zinov’ev and Lebedev, 1976); 2—drop method (Arpaci and Frohberg, 1984); 3, 4—pulse calorimetry (Yakunkin, 1983; Senchenko and Sheindlin, 1987); 5—dynamic calorimetry (Righini et al., 1993).
was employed in studies of other high-melting-point metals: tantalum (Kraftmakher, 1963a), niobium (Kraftmakher, 1963b), molybdenum (Kraftmakher, 1964), and platinum (Kraftmakher and Lanina, 1965). These data are presented in Table 5.1. The nonlinear increase in the speciFc heat of all the metals was treated as being the result of point-defect formation in the crystal lattice (Kraftmakher, 1966c). The equivalent-impedance technique reduces measurements of the enthalpy to determinations of the resistance of the sample at given temperatures: T2 R2 H2 − H1 = mc dT = (2I02 =!2 RC) dR ; (5.6) T1
R1
where R1 and R2 are the resistances at the temperatures T1 and T2 . When the thickness of the sample is of the order of 1 mm, the temperature derivative of its resistance can be measured using a thin thermocouple welded to the sample. A DC current and an AC current modulated by an infralow frequency pass through the sample. Temperature oscillations with a period of several seconds and corresponding oscillations of the resistance thus occur in the sample. The voltage drop across a central portion of the sample contains a component of the infralow frequency. It is proportional to the amplitude of the temperature oscillations, to the DC current, and to R , the temperature derivative of the sample’s resistance. A two-channel recorder or X–Y plotter records the oscillations of the thermocouple’s voltage and of the voltage drop across the central portion of the sample. The adiabatic regime is not required in such measurements. The modulation frequency may therefore be reduced to provide signiFcant temperature oscillations and to reduce the inEuence of the thermal inertia of the thermocouple. The data on R derived in this way are then used in measurements of the speciFc heat at higher frequencies corresponding to the adiabatic regime. A moderate vacuum suIces for the measurements at middle temperatures to avoid instabilities due to convection. However, a perfect vacuum is needed in high-temperature studies, and the best method to obtain it is cryopumping. The most diIcult case is when an inert gas atmosphere is used.
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Table 5.1 SpeciFc heat measured by equivalent-impedance method (smoothed values, J mol−1 K −1 ) T (K) 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600
W
28.7 29.2 29.75 30.3 30.85 31.4 32.0 32.6 33.2 33.9 34.65 35.5 36.55 37.8 39.3 41.05 43.25 45.9 49.05 52.8 57.2 62.4
Ta
27.75 28.05 28.35 28.65 28.9 29.2 29.5 29.8 30.1 30.4 30.7 31.1 31.5 31.95 32.5 33.2 34.05 35.0
Mo
30.5 31.15 31.8 32.5 33.25 34.05 34.95 36.0 37.3 38.9 40.9 43.4 46.5
Nb
27.5 28.05 28.6 29.15 29.7 30.3 31.0 31.75 32.55 33.55 34.65 35.95 37.45 39.2 41.15
Pt 29.95 30.5 31.05 31.65 32.35 33.15 34.1 35.3 36.8 38.7 41.05
5.2.2. Nonadiabatic regime The adiabatic regime of the measurements is achievable by increasing the modulation frequency. However, in some cases such an increase is undesirable (e.g., when the heat is generated at the surface of the sample or when a thermometer used has a long time constant). In a nonadiabatic regime, one has to determine both the amplitude and the phase of the temperature oscillations, and the speciFc heat is given by the relation (2.6). It was shown that measurements in a nonadiabatic regime are possible with about the same accuracy as in the adiabatic regime (Varchenko and Kraftmakher, 1973). Although the heat losses under nonadiabatic conditions are signiFcant, they are readily taken into account. Moreover, in a nonadiabatic regime one can measure the speciFc heat using the heat transfer coeIcient as a reference quantity. In the case of the radiative heat transfer, this coeIcient is governed by the temperature dependence of the hemispherical total emittance of the sample.
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5.2.3. Direct measurement of the temperature coe?cient of the speci@c heat The direct determination of the temperature coeIcient of the speciFc heat (TCSH) rests on a strictly sinusoidal modulation of the heating power. The temperature dependence of the speciFc heat brings about a deviation of the temperature oscillations from the sinusoidal form. A second-harmonic component in the temperature oscillations appears, which depends on TCSH (Kraftmakher and Tonaevskii, 1972). The heat-balance equation, with the temperature dependence of the speciFc heat taken into account, is mc(1 + #) + P + P + P 2 =2 = p0 + p cos !t ; P,
(5.7)
P
where # = (1=c) dc=dT is the TCSH, and P, and are the power of heat losses from the sample and its temperature derivatives. Since #1, this nonlinear equation is solvable by successive approximations. With high-order terms neglected, the solution has the form = 1 cos(!t − 1 ) + 2 cos(2!t − 2 ) ;
(5.8a)
1 = (p sin 1 )=mc! ;
(5.8b)
2 = #21 =4 cos 2 ;
(5.8c)
tan 1 = mc!=P ;
(5.8d)
tan 2 = P =2mc!# :
(5.8e)
∼ 90 , 2 = ∼ 0, and the above expressions become At suIciently high modulation frequencies, 1 = simpler: ◦
1 = p=mc! ;
(5.9a)
2 = #21 =4 :
(5.9b)
The TCSH is thus available from the fundamental component and the second harmonic of the temperature oscillations. Above the Debye temperature, # is usually of the order of 10−4 K −1 . It may increase several fold due to the point-defect contribution. Close to a second-order phase transition, # is in the range 10−2 –10−1 K −1 , and one may succeed in measuring the second harmonic even with temperature oscillations smaller than 1 K. In all such cases, a high selectivity is necessary to accurately measure the second harmonic in the presence of a much stronger fundamental signal. A strictly sinusoidal modulation of the heating power is diIcult to attain in practice. It is convenient to heat wire samples by an AC current. When the internal resistance of the source is suIciently high for all harmonics, no changes of the current are caused by the oscillations of the sample’s resistance. However, a second harmonic in the heating-power oscillations arises because the current Eows through an oscillating resistance. To take this into account, the right-hand side of Eq. (5.7) should be written as I 2 R(1 + ) cos2 (!t=2), where I is the amplitude of the heating current, R is the sample’s resistance, !=2 denotes the frequency of the current, and
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= R =R is the temperature coeIcient of resistance. Since 1, one obtains, instead of (5.9b), 2 = 21 (# − )=4 : (5.10) Of great importance is the method of measuring the temperature oscillations. If they are detected through a parameter nonlinearly depending on temperature, the results may be markedly distorted. For example, the temperature oscillations should not be measured through the radiation from the sample. It is better to use the temperature dependence of the sample’s resistance. To calculate the corrections, the increment in the resistance should be expanded in a series with the Frst two terms retained: TR = R + R 2 =2 ; (5.11) where = d =dT . The temperature oscillations at the fundamental frequency result therefore in the resistance oscillations also at the doubled frequency. Finally, 2 = 21 (# − − = )=4 : (5.12) Hence, when heating the sample by an AC current and measuring the temperature oscillations through its electrical resistance, Eq. (5.12) is no longer as simple as (5.9b). However, such a method is quite acceptable in many cases. As an example, the quantities entering Eq. (5.12) were calculated for tungsten (Kraftmakher and Tonaevskii, 1972). When approaching the melting point, the main part is played by # due to the strong nonlinear increase in the speciFc heat, while prevails at low and middle temperatures. The correction associated with = remains relatively small. Two gas-Flled incandescent lamps with tungsten Flaments served as the samples to test the method. They formed a part of a bridge fed by a 50 Hz mains current (Fig. 5.7). This frequency is suIciently high to ensure the validity of Eqs. (5:9). Since the expected 2 =1 ratio is 10−3 –10−2 , the harmonic content in the heating current must be small. The harmonics of the temperature oscillations are measured in the following way. A small supplementary current of a frequency close to that of the harmonic to be measured passes through the samples. This results in the appearance of a voltage drop across the sample at the di0erence frequency, of about 0:1 Hz. This voltage is proportional to the magnitude of the oscillations in the sample’s resistance and to the supplementary current. It is fed through an RC Flter to an ampliFer and then recorded by a plotter. By tuning the frequencies of the supplementary current close to ! and 2!, one in turn measures the components of the temperature oscillations. The method is capable of measuring the second harmonic in the presence of a much stronger fundamental signal (Fig. 5.8). The strong increase of TCSH corresponds well to the nonlinear increase of the speciFc heat. It is not evident whether this method is applicable for studying second-order phase transitions. In this case, the temperature oscillations must be kept small to ensure good temperature resolution. A thermocouple is the most proper tool to measure the temperature oscillations in such studies. 5.2.4. Bulk samples Electron-bombardment heating is a method suitable for bulk samples or samples of irregular shape. The temperature oscillations in the sample are measured through the radiation from it
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Fig. 5.7. Direct measurement of the temperature coeIcient of the speciFc heat (Kraftmakher and Tonaevskii, 1972). The low-frequency oscillator is in turn tuned close to the frequency of the temperature oscillations and to its second harmonic. Fig. 5.8. Temperature dependence of 2 =1 ratio for tungsten. Negative values below 3200 K show predominant contribution of (Kraftmakher and Tonaevskii, 1972).
or by means of a thermocouple. This method was used in measurements of the speciFc heat of iron near its Curie point (Varchenko et al., 1978). A blackbody model is necessary to reliably measure temperature oscillations in the sample through radiation from it. 5.2.5. Molten metals To measure the speciFc heat of molten metals, Akhmatova (1965, 1967) Flled a niobium capillary with them. The capillary was located in a vacuum chamber and heated by passing an AC current through it. The method rests on the fact that the temperature dependence of the radiation from the empty and the Flled capillary is the same. This allows one to directly compare the temperature oscillations at a given mean temperature. The oscillations are inversely proportional to the total heat capacity of the capillary, empty or Flled with the melt. The heat capacity of the melt thus is Ftted to that of the capillary. The mean temperature is determined by an optical pyrometer or from the electrical resistance of the empty capillary. A photomultiplier detects the temperature oscillations. An advantage of this approach is that one has no need to know absolute values of the temperature oscillations. The capillary serves as a calorimeter and a reference of the speciFc heat. This method seems to be the simplest one and quite adequate. Using it, Akhmatova (1965, 1967) determined the speciFc heat of molten gallium, tin, and copper. 5.2.6. Active thermal shield The heat transfer coeIcient grows rapidly with temperature. It is proportional to T 3 when the heat losses are due to radiation from the sample. In fact, the increase is somewhat stronger because of the temperature dependence of the hemispherical total emittance. As a rule, it
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Fig. 5.9. Compensation for heat losses by means of an active thermal shield (Kraftmakher and Cherepanov, 1978). Temperature oscillations in the shield are adjusted to make temperature oscillations in the sample to be lagged by ◦ 90 to the oscillations in the heating power.
increases with temperature. Therefore, the modulation frequency required to meet adiabatic conditions grows along with the temperature. Low modulation frequencies can be maintained using a nonadiabatic regime. The second possibility is compensation for heat losses from the sample by means of an active thermal shield (Kraftmakher and Cherepanov, 1978). A sample in a vacuum chamber is surrounded by a shield whose temperature oscillates with the same frequency and phase as that of the sample (Fig. 5.9). The amplitude of the temperature oscillations in the shield is adjusted to nullify the AC component of the heat losses from the sample. ◦ The criterion of the compensation is a 90 phase shift between the oscillations of the heating power and of the sample’s temperature. The heat-balance equation now takes the form mc + (K1 − bK2 ) = p sin !t :
(5.13)
Here K1 and K2 are the temperature derivatives of the mutual heat transfer coeIcients for the sample and the shield, and b is the ratio of the amplitudes of the temperature oscillations in the shield and in the sample. The solution to this equation is mc = (p=!0 ) sin ;
(5.14a)
tan = mc!=(K1 − bK2 ) :
(5.14b)
The criterion of adiabaticity, tan 1, thus is satisFed at low frequencies by adjusting the amplitude of the temperature oscillations in the shield. 5.2.7. Nonconducting materials To measure the speciFc heat of nonconducting materials, one has to satisfy some additional requirements. The conditions of adiabaticity and of thermal equilibration in the calorimeter pose contradictory requirements for the modulation frequency. At low and middle temperatures, when the heat transfer coeIcient is small, it is easy to Fnd a modulation frequency that meets both requirements. Thin samples, of several tenths of a millimeter, ensure internal thermal equilibration. Many modulation measurements on nonconducting samples were carried out using modulated-light heating. As a rule, no determinations of absolute magnitudes of the heating-power oscillations were made. The resolution of such measurements in narrow
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temperature intervals is of the order of 0.01%, but the inaccuracy of the absolute values may amount to 5 –10%. Glass (1968) was the Frst to apply modulation calorimetry to a nonconducting material. He determined the speciFc heat of LiTaO3 in the range 300 –1000 K. A sample 0:1 mm thick was placed in a furnace and illuminated by chopped light from an incandescent lamp. The modulation frequency was 30 Hz. A thermocouple and a lock-in ampliFer detected the temperature oscillations in the sample. A photocell provided the reference voltage. The output voltage of the lock-in ampliFer was fed to the Y-input of a plotter, while a thermocouple measuring the mean temperature in the furnace was connected to the X-input. Hence, the setup was similar to that developed by Handler et al. (1967). A separate heater was employed in measurements on LiKSO4 in the range 400 –750 K (Breczewski ; et al., 1984). A thin nickel Flm on one side of the sample served as the heater. A thermocouple junction was attached to the backside of the sample. Higher temperatures, up to 1500 K, were achieved in measurements on some melts (Derman and Bogorodskii, 1970). The liquid under study Flled a hollow platinum crucible placed in a furnace. A heater of a low thermal inertia passed along the axis of the crucible. The setup served to create radial temperature waves in the crucible. However, over a deFnite frequency range the amplitude of the temperature oscillations depended only on the heat capacity of the sample. The uncertainty in the speciFc-heat data amounted to 4%. One can also place a Eat resistive heater between two portions of a sample and attach a thermocouple to the outer side of the sample (Filippov and Yurchak, 1965). The equivalent-impedance technique is capable of measuring the speciFc heat of a thin nonconducting layer deposited on a conducting wire or strip. A metal of known speciFc heat, tungsten or platinum, may serve as the main sample. In this case, one directly compares the heat capacities of the coating and of the main sample. Moreover, there is no need to know the temperature derivative of the sample’s resistance because it is the same for the sample with and without the nonconducting coating. The main sample thus serves as a heater, a thermometer, and a reference of speciFc heat. Pochapsky (1953) successfully employed this method in pulse-heating measurements of the speciFc heat of AgBr deposited on a platinum wire. The nonconducting layer must be of constant thickness and good adhesion to the main sample must be ensured. A relation for evaluating the speciFc heat of the layer follows from the basic expression for the equivalent-impedance method. Under adiabatic conditions, Eq. (4.8) is valid for the main sample. For the composite sample, including the coating of heat capacity m1 c1 , the corresponding relation is 2 mc + m1 c1 = 2I01 R =!2 RC1 :
(5.15)
At a given temperature, the values of R and R are the same in both measurements, as well as the modulation frequency. Only the DC current heating the sample and the capacitance corresponding to the equivalent impedance of the sample alter. One thus obtains 2 m1 c1 =mc = I01 C=I02 C1 − 1 :
(5.16)
A similar expression can be derived for a nonadiabatic regime. A nonadiabatic regime allows one to use the phase shift between the heating-power oscillations and the temperature oscillations, instead of the amplitude of the temperature oscillations. For instance, one may
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Fig. 5.10. Variants of modulation measurements on nonconducting materials: (a) plain heater between two portions of the sample; (b) heating by radiation or electron bombardment; (c) sample in a metal capillary heated by electric current; (d) sample in a crucible heated by electron bombardment.
employ a conducting capillary of known heat capacity heated by a DC current with a small AC component. When the mean temperature of the capillary varies, the phase shift is kept ◦ constant, e.g., 45 , by adjusting the modulation frequency. For this phase shift, tan = 1, and ! = P =mc at every mean temperature. Then the capillary is Flled with the sample and the run repeated. The temperature dependence of the heat transfer coeIcient P remains the same. The new temperature dependence of the frequency !1 necessary to maintain the same phase shift depends on the heat capacity of the sample, m1 c1 . Clearly, !=!1 = (mc + m1 c1 )=mc :
(5.17)
The capillary serves here as a heater and a reference of speciFc heat, and there is no need to calibrate the sensor measuring the temperature oscillations. To measure the speciFc heat of nonconducting materials, it is enough to prepare samples of suitable shape compatible with the container and the heater at high temperatures (Fig. 5.10). Table 5.2 lists modulation measurements carried out at temperatures above 1000 K. 5.3. Low temperatures At low temperatures, oscillations in the power heating the sample are readily measurable when using direct electric heating or separate heaters. Resistance thermometers or thermocouples serve to measure the mean temperature of the sample and the temperature oscillations in it. Platinum thermometers are usable down to 10 K, whereas a semiconductor thermometer is necessary for lower temperatures. The sensitivity of the latter increases with decreasing temperature. Using lock-in detection, very high sensitivity is achievable. For simplicity, many workers employ modulated-light heating. Usually, the front surface of the sample is painted black to enhance and stabilize the absorption of light, which thus becomes independent of temperature. The advantages of modulation calorimetry are especially signiFcant when speciFc heat is measured, at a given temperature, as a function of an external parameter such as a magnetic
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Table 5.2 Modulation calorimetry at high temperatures (T ¿ 1000 K) Item
Reference
W, via thermionic emission W, up to 2600 K W, 1500 –3600 K Nb, Mo, Ta, W, 1200 –2400 K, photosensor Ta, 1200 –2900 K Nb, 1300 –2700 K Mo, 1300 –2500 K Molten Sn, Ga and Cu, photosensor Ge, Si, GeSi alloys Pt, 1000 –2000 K Graphite, 1750 –2850 K, photosensor Ti, 1400 –1900 K, photosensor Zr, 1300 –2000 K, photosensor Au, 700 –1300 K W, 1300 –3000 K Cu, 550 –1250 K, thermocouple WRe, 1600 –2900 K, photosensor Mo, 1100 –2400 K, photosensor Nb, 1100 –2400 K, photosensor Au, 470 –1220 K Ir, 1500 –2500 K, photosensor W, temperature coeIcient of speciFc heat Pt, 1200 –1900 K, search for relaxation Au, relaxation phenomenon at 1064 K W, 800 –2500 K, via thermal noise WRe, 1000 –2700 K, via thermal noise Pt, active thermal shield W, Cp =Cv via temperature Euctuations W, relaxation in speciFc heat Pt and PtRh, thermocouple Ni, 900 –1400 K Pt, relaxation in speciFc heat Nb, noncontact measurements
Smith and Bigler (1922), Bockstahler (1925) Zwikker (1928) Kraftmakher and Strelkov (1962) Lowenthal (1963) Kraftmakher (1963a) Kraftmakher (1963b) Kraftmakher (1964) Akhmatova (1965, 1967) Gerlich et al. (1965) Kraftmakher and Lanina (1965) Kraftmakher and Shestopal (1965) Shestopal (1965) Kanel’ and Kraftmakher (1966) Kraftmakher and Strelkov (1966a) Kraev (1967) Kraftmakher (1967c) Sukhovei (1967) Makarenko et al. (1970a) Makarenko et al. (1970b) Skelskey and Van den Sype (1970) Trukhanova and Filippov (1970) Kraftmakher and Tonaevskii (1972) Seville (1974) Skelskey and Van den Sype (1974) Kraftmakher and Cherevko (1974) Kraftmakher and Cherevko (1975) Kraftmakher and Cherepanov (1978) Kraftmakher and Krylov (1980a, b) Kraftmakher (1985) Glazkov and Kraftmakher (1986) Glazkov (1987) Kraftmakher (1990) Wunderlich and Fecht (1993), Wunderlich et al. (1993) Wunderlich and Fecht (1996) Wunderlich et al. (1997), Egry (2000)
ZrNi36 ; ZrFe24 ; ZrCo23:5 Noncontact calorimetry in space
Feld or pressure. When the sensitivity of the thermometer employed does not depend on the magnetic Feld, one immediately obtains the Feld dependence of the speciFc heat. Sullivan and Seidel (1966, 1967, 1968) have shown this important feature in the Frst modulation measurements at low temperatures. Quantum oscillations in the speciFc heat were observed in beryllium, as well as the destruction of superconductivity in indium. In these measurements, the temperature oscillations in the samples were in the range 2–3 mK. At 1 K and 30 s averaging time, the sensitivity to changes in the temperature oscillations was nearly 10−7 K. The imprecision
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Table 5.3 Modulation calorimetry at low temperaturesa Item
Reference
Be, 1:5 K, in magnetic Feld Pb–Ag–Pb sandwiches, 0.3–1:2 K TTF-TCNQ Au1−x Nix , amorphous, 2–40 K Pb70 Bi30 , 0.5 –2 K As, amorphous, 1.8–20 K Au1−x Fex , amorphous, 1.5 –18 K Si, amorphous, 2–50 K Al80 Mn20 ; Al78 Mn22 , 0.5 –5 K GaAs=Alx Ga1−x As, 2-D electron gas UBe13 , 0.3–1:8 K, in magnetic Feld up to 20 T #-(BEDT-TTF)2 I3 Ne, Ar, Kr, Xe, Ne–Xe, Ar–Xe Flms, 0.1–7 K Mx TiS2 intercalates (M = Mn; Fe; Co; Ni) Graphite-ICl intercalate, 2.2–5 K (TMTSF)2 ReO4 ; (TMTSF)2 BF4 (-(BEDT-TTF)2 Cu[N(CN)2 ]Cl; (TMTSF)2 ReO4 #-(BEDT-TTF)2 MHg(NCS)4 ; M = K; Rb TSeF-TCNQ 4 He, superEuid transition
Sullivan and Seidel (1967) Manuel and VeyssiRe (1972) Craven et al. (1974) Eno et al. (1977) K[ampf and Buckel (1977) Lannin et al. (1978) Dawes and Coles (1979) Mertig et al. (1984) Machado et al. (1987a, b) Wang et al. (1988, 1992) Graf et al. (1989) Fortune et al. (1991) Menges and von L[ohneysen (1991) Inoue et al. (1986), Takase et al. (1994) Tashiro et al. (1990) Chung et al. (1993a) Chung et al. (1993b) Henning et al. (1995) Powell et al. (1997) Mehta and Gasparini (1997, 1998), Mehta et al. (1999) Fominaya et al. (1997b, 1999b) Akutsu et al. (1999) Fominaya et al. (1999a) Akutsu et al. (2000)
Mn12 acetate, in magnetic Feld (DMET)2 BF4 ; (DMET)2 ClO4 Fe8 crystal, 1.5 –10 K, in magnetic Feld (-(BEDT-TTF)2 Cu[N(CN)2 ]X; X = Br; Cl
a Fe8 crystal = [(triazacyclononane)6 Fe8 O2 (OH)12 ]8+ ; TTF-TCNQ = tetrathiafulvalene-tetracyanoquinodimethane, TMTSF = tetramethyltetraselenafulvalene, BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene, DMET = dimethyl (ethylenedithio)diselenadithiafulvalene.
of the measurements was better than 0.1%. Later, many investigators measured speciFc heat versus external magnetic Feld (e.g., Zoller and Dillinger, 1969; Suzuki and Tsuboi, 1977; Shang et al., 1978; Shang and Salamon, 1980; Fortune et al., 1990; Izawa et al., 1996; Fominaya et al., 1997b, 1999a, b). Table 5.3 lists some modulation measurements carried out at low temperatures. Further examples are given in Section 11.2.3. 5.4. Measurements under high pressures Chu and Knapp (1973) reported the Frst modulation measurements under high pressures, up to 1:2 GPa. They determined the phase diagram of an #-U crystal at low temperatures. Nb3 Sn and V3 Si also were investigated (Chu, 1974; Chu and Testardi, 1974; Chu and Vieland, 1974). Bonilla and Garland (1974) determined the speciFc heat of chromium along its NReel line, up to 0:3 GPa.
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Fig. 5.11. Presentation of calorimetric cell under hydrostatic gas pressure (Baloga and Garland, 1977).
Baloga and Garland (1977) examined modulation calorimetry under high pressures with a gas as the pressure-transmitting medium. The sample, a slab of dimension 10 × 10 × 0:5 mm3 , with its addenda is suspended by leads in a cylindrical copper sample holder (Fig. 5.11). A thin Flm of resistance epoxy painted directly onto the face of the sample serves as the heater. A microbead thermistor is cemented to the opposite face. The main conclusions by the authors are as follows: (i) the sample must have large faces to exclude edge e0ects on the temperature at the center; (ii) the sample must be thin enough to avoid signiFcant temperature di0erences during a period compatible with a reasonable modulation frequency; (iii) it is desirable to minimize heat leaks from the sample through the heater and thermistor leads; (iv) the heat capacity of the addenda should be small compared to that of the sample; and (v) the heater and thermistor should be electrically and mechanically stable on cycling pressure and temperature. The authors measured the speciFc heat of NH4 Cl and ND4 Cl up to 0:3 GPa (Garland and Baloga, 1977). Eichler and Gey (1979) developed a modulation calorimeter for measurements under high pressures at low temperatures. Diamond powder served as a pressure-transmitting medium. At low temperatures, it has a low speciFc heat and a low thermal conductivity. This reduces the contribution of the medium and corrections for heat losses. The sample geometry was chosen to compromise between the requirements of calorimetric and high-pressure techniques. An indium sample consisted of three discs, 3 mm in diameter and 0.5 –1:5 mm thick. The heater and thermometer were each mounted between two discs. The heater was made of 30-m manganin wire, and the thermometer was cut from a standard Allen–Bradley carbon resistor. A circular slice, 1 mm in diameter, was ground and thoroughly polished from both sides to a thickness of about 0:15 mm. The whole assembly consisting of the indium discs and the embodied measuring elements was placed in a hollow cylinder of compressed diamond powder, with a grain size smaller than 0:5 m. The upper and lower covers of this cylinder were diamond discs. The surrounding BeCu pressure cylinder served as the temperature bath. The pressure cell was put into a device capable of changing the pressure at any temperature between 1.3 and 300 K. The apparatus contained in a chamber Flled with a small amount of helium exchange gas was placed in the bore of a superconducting magnet. The external and internal time constants at 4:2 K; 0:5 s and 0:5 ms, are quite favorable to fulFll the criterion of adiabaticity. The operating frequency, 20 Hz, was near the upper limit of the frequency band where the quantity !0 remains constant at 4:2 K. The lower limit was nearly 2 Hz. Lock-in ampliFers measured all
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Table 5.4 Measurements under high pressuresa Item
Reference
#-U, 10 –40 K Cr, NReel line Nb3 Sn V3 Si, speciFc heat and dR=dT Nb3 Sn, speciFc heat and dR=dT Cs, liquid, up to 1700 K NH4 Cl; ND4 Cl
Chu and Knapp (1973) Bonilla and Garland (1974) Chu (1974) Chu and Testardi (1974) Chu and Vieland (1974) Filippov et al. (1976) Baloga and Garland (1977), Garland and Baloga (1977) Itskevich et al. (1978) Eichler and Gey (1979), Eichler et al. (1981) Garland et al. (1979), Kasting et al. (1980) Eichler et al. (1980) Polandov et al. (1981) Stokka and Fossheim (1982b) Gulish et al. (1983) Jin et al. (1984) Blagonravov et al. (1983) Blagonravov et al. (1984) Bleckwedel and Eichler (1985) Bohn and Eichler (1991) Kirsch et al. (1992) Chen et al. (1993) Leyser et al. (1995) Bouquet et al. (2000)
Sn, superconductor In, superconductor 8OCB, liquid crystal Ga, superconductor TGSe KMnF3 TGS + TGSe mixtures CeSb Cs, liquid, up to 2000 K Rb, liquid, up to 1900 K CeCu2 Si2 La, low temperatures CrPb3 Eu0:9 Ho0:1 Mo6 S8 , magnetic Felds up to 20 T o-terphenyl, glass transition, 2–6300 Hz CeRu2 Ge2 , 1.5 –11 K a
8OCB = octyloxycyanobiphenyl; TGSe = triglycine selenate, TGS = triglycine sulfate.
the AC voltages. A data-acquisition system controlled the measurements. The system measured each quantity repeatedly, 20 – 60 times, and provided the mean values. The total error of the measurements was 3%. There is no way to experimentally determine the heat capacity of the addenda consisting of the heater, the thermometer, and certain proportions of the electric leads and the surrounding diamond powder. In the temperature range 1:3–5 K, this contribution was estimated to be less than 1%. Filippov et al. (1976) and Blagonravov et al. (1983, 1984) performed the Frst modulation measurements under high pressures and high temperatures. Table 5.4 lists measurements carried out at high pressures. 5.5. Modulation microcalorimetry 5.5.1. Thin deposited @lms The high resolution peculiar to modulation calorimetry makes it possible to perform measurements on small samples whose heat capacity is much smaller than that of the addenda. Zally and Mochel (1971, 1972) measured the low-temperature speciFc heat of thin superconducting BiSb Flms deposited onto a substrate (Fig. 5.12). A disc of 2–3 m Pyrex glass is mounted on
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Fig. 5.12. Microcalorimetry of thin Flms at low temperatures (Zally and Mochel, 1971, 1972).
Fig. 5.13. Measurement of speciFc heat of thin free-standing liquid-crystal Flms (Pitchford et al., 1986).
a copper ring suspended inside a helium-bath shield. A 10-nm constantan heater is evaporated onto the glass. A 500-nm layer of silicon monoxide evaporated on top of the heater acts as an insulator. A 300-nm dot of indium evaporated on the opposite side of the glass improves thermal di0usion over the region to be occupied by the sample. A thermometer made out of a small chip of Sb-doped germanium is sealed to the indium. At 2:2 K, a typical transition temperature, the heat capacity of the substrate assembly was about 10−7 J K −1 . For a 100-nm BiSb Flm, the heat-capacity discontinuity at the transition point was of the order of 10−9 J K −1 . The amplitude of the temperature oscillations was in the range 2–5 mK, while the resolution was better than 10−3 . Manuel and VeyssiRe (1976), K[ampf and Buckel (1977), K[ampf et al. (1981), Rao and Goldman (1981), Suzuki et al. (1982) also studied thin Flms at low temperatures. 5.5.2. Free-standing @lms Pitchford et al. (1986) have found an unusual approach to measuring the speciFc heat of thin free-standing liquid-crystal Flms (Fig. 5.13). A free-standing Flm contains 10 –1000 smectic layers. The Flm is kept inside a regulated oven Flled with argon gas at 50 kPa. Chopped radiation from a He–Ne laser with 1-mW output power at ) = 3:4 m falls onto the Flm. A small thermocouple located beneath the Flm measures the temperature oscillations in it. The separation between the thermocouple junction and the Flm is about 0:1 mm. At the modulation frequency 5:6 Hz, it is much smaller than the thermal di0usion length in the gas, 1:2 mm. A thermistor located beneath the Flm but away from the thermocouple junction measures the mean temperature. The measurements have shown that the radiation absorbed by the Flm is independent of temperature. This method allowed the authors to measure the speciFc heat of Flms ranging from three to a few thousand smectic layers (Geer et al., 1989, 1991a, b).
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Fig. 5.14. Bath-modulation calorimetry (Graebner, 1989).
Stoebe et al. (1992) and Jin et al. (1995) studied the phase transition in a free-standing Flm containing only two layers. 5.5.3. Bath modulation This method was proposed by Graebner (1989) and employed in studies of small superconducting samples (Fig. 5.14). A small sample is pasted to a thermocouple junction. Short wires of the thermocouple are attached to a light platform whose temperature is modulated. The thermocouple wires provide the oscillating heat input for the sample. In a certain frequency range, the AC component of the thermocouple response depends on the heat capacity of the sample. The mass of the sample was only 2:9 g. The method was used in measurements of the speciFc heat of superconductors versus magnetic Feld, up to 6 T (Graebner et al., 1989, 1990). 5.5.4. Nanocalorimetry at low temperatures SigniFcant e0orts were applied to decrease the size of samples acceptable for modulation calorimetry. Campbell and Bretz (1985) and Kenny and Richards (1990a) achieved sensitivity at low temperatures of the order of 10−10 J K −1 . Chae and Bretz (1989) improved it to 10−11 J K −1 . Phelps et al. (1993) and Birmingham et al. (1996) also performed modulation measurements on small samples. Riou et al. (1997) designed a high-resolution microcalorimeter optimized for the range 40– 160 K. The sample holder is made out of a polyphenylquinoxaline (PPQ) membrane suspended on a copper round disc (Fig. 5.15). The advantage of PPQ is its high thermal stability, good adhesion on copper, and high elasticity. A gold thin Flm of 0.6-mm diameter sputtered in the center of the membrane deFnes an isothermal area. The lithographic elements (DC and AC heaters and a copper thermometer) sputtered on the other side of the membrane are connected to the copper disc through narrow metallic pads. The copper disc is coupled via a calibrated thermal link to a copper plate at liquid helium temperature. An electric heater and two thermometers, a standard platinum thermometer and a germanium thermometer, are mounted on the copper disc. The temperature of the disc is kept constant at 10:15 K. A DC power is supplied to the DC heater of the membrane to achieve the desired mean temperature. Then an AC power is supplied
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Fig. 5.15. Modulation microcalorimeter designed by Riou et al. (1997). Fig. 5.16. Modulation nanocalorimeter designed by Fominaya et al. (1997a).
to the AC heater at the center of the membrane. The mass of a sample is typically 10 g. At 100 K, the speciFc heat of the addendum is 1:5 × 10−6 J K −1 . The temperature oscillations, in the range 5–70 mK, are measured with a lock-in ampliFer. The averaging time varies from 30 s to 3 min. With this calorimeter, Charalambous et al. (1999) investigated the critical behavior of an YBCO single crystal. Fominaya et al. (1997a) further developed modulation microcalorimetry. A low-temperature calorimeter has been designed for measurements on thin Flms and small single crystals. The heat capacity of the addenda was 3 × 10−9 J K −1 at 4 K and 5 × 10−10 J K −1 at 1:5 K. The resolution of the order of 10−4 makes it possible to see variations in speciFc heat smaller than 10−12 J K −1 . This achievement o0ered a new Feld in the modulation technique, modulation nanocalorimetry. In the fabrication of the nanocalorimeter (Fominaya et al., 1997a), the approach was similar to that taken in designing electronic microchips, and the corresponding technology was employed (Fig. 5.16). A silicon substrate (10 × 15 × 0:28 mm3 ) presents a base for the calorimeter. A Si3 N4 protection layer, 200 nm thick, was removed from a square area of 5 × 5 mm2 by means of SF6 plasma reactive ion etching. This area was further etched in KOH to obtain a 5-m silicon membrane suspended on a thick frame. Then a 100-nm NbTi Flm and a 30-nm platinum Flm were deposited by magnetron sputtering to form leads and pads. Optical lithography, ion beam etching, and chemical etching served for this purpose. Further, a sputtered 150-nm CuNi layer formed three heaters, one on the membrane and two on the silicon frame. Then a 150-nm NbN Flm was sputtered and patterned to create thermometers. To improve the thermal isolation of the membrane, holes were etched with a SF6 plasma into the membrane, so that a (3:3 × 3:3)-mm2 membrane remained, being suspended by 12 40-m bridges. The thermal coupling of the membrane to the frame thus depends on the geometry of the bridges and the doping of the silicon. The sample was grown on the membrane, between the heater and the thermometer. Second, one can paste a small single crystal on the backside of the membrane. The silicon frame is anchored to a copper holder provided with a germanium thermometer and linked to the bath. The whole is installed in a conventional low-temperature calorimeter. A 5-T superconducting coil
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provides the magnetic Feld. A low-noise preampliFer and a lock-in ampliFer measure the temperature oscillations. A heater governed by the germanium thermometer on the holder or by the NbN thermometer on the frame regulates the temperature. The feasibility of the nanocalorimeter was conFrmed by measurements on a thin lead layer and on single crystals of Mn12 acetate. In the 150-nm lead Flm, a deviation from bulk behavior was found under external magnetic Felds. In a magnetic Feld, the speciFc-heat curve near the transition point has a shape similar to that in type II superconductors. The reason is that the Flm’s thickness is smaller than the coherence length, so that a magnetic Eux penetrates the sample. The speciFc heat of Mn12 acetate was measured, at several temperatures, as a function of magnetic Feld (Fominaya et al., 1997b, 1999b). Fominaya et al. (1999a) carried out a similar study on a Fe8 single crystal, [(triazacyclononane)6 Fe8 O2 (OH)12 ]8+ . 5.6. Photoacoustic technique The photoacoustic phenomenon was known for about a century before it was applied to modulation measurements of the speciFc heat and other thermophysical properties of solids. Rosencwaig and Gersho (1976) have developed a theory of such measurements. The authors claimed that the main source of the acoustic signal in a photoacoustic cell arises from the periodic heat Eow from the solid to the surrounding gas when the solid is periodically heated. Only a relatively thin layer of air adjacent to the surface of the solid participates in the production of the acoustic signal. The general solution for the thermal-di0usion equations contains many parameters related to the solid, the backing, and the gas. However, for special cases that are readily achievable, the solution becomes much simpler (for details see Harren and Reuss, 1997). Zammit et al. (1988) described an experimental setup for the photoacoustic calorimetry. With this technique, Zammit et al. (1990), Marinelli et al. (1992), Schoubs et al. (1994) studied phase transitions in liquid crystals. Using the photoacoustic and photopyroelectric techniques, Glorieux et al. (1994) investigated antiferromagnetic phase transitions in CoO and Cr 2 O3 . At low modulation frequencies, the thermal di0usivity, D = )=*c, and the e0usivity, e = (*c))1=2 , govern the amplitude and the phase of the signal. Here ) is the thermal conductivity and * is the density of the sample. At high frequencies, the signal becomes inversely proportional to the e0usivity of the sample. By combining the data, the speciFc heat and the thermal conductivity are available. Glorieux and Thoen (1994) and Glorieux et al. (1995) used the photoacoustic technique to simultaneously determine the speciFc heat and the thermal conductivity of gadolinium near its Curie point. The mean temperature of the photoacoustic cell was automatically controlled within 0:01 K, and the heating rate was 0:02 K min−1 . A modulated beam from a 10-mW He–Ne laser illuminated the sample. A microphone followed by a lock-in ampliFer detected the photoacoustic signal. The amplitude and the phase of the temperature oscillations are available from a calibration by means of a reference sample with known optical and thermal properties, e.g., carbon-coated copper. Through iterative calculations, the signal provided the e0usuvity and the thermal di0usivity of the sample, from which the speciFc heat and the thermal conductivity were determined. In the range 10–240 Hz both quantities were fre◦ quency independent. Fitting the data to reference values at 40 C yielded absolute values.
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The measurements clariFed the behavior of gadolinium samples of di0erent quality in magnetic Felds. It turned out that employment of high-quality single-crystal samples avoids rounding e0ects. In addition, the temperature oscillations in the samples were created also by an AC magnetic Feld. 5.7. Noncontact calorimetry Noncontact modulation calorimetry involves electromagnetic levitation and optical pyrometry. The heating power is supplied to the sample by means of a laser beam or RF heating. This technique allows one to measure the speciFc heat of small solid particles and supercooled liquids. Monazam et al. (1989) employed an electrodynamic balance for contactless measurements of absorptivities and heat capacities of single particles, 0:05–0:2 mm in diameter, kept in a vacuum chamber. The setup included a 50-W CO2 laser and the optics needed to direct the beam into the chamber. The laser provided 3-ms heating pulses at a 100-Hz repetition rate. A high-speed optical pyrometer monitored the resulting periodic temperature changes in the samples. The authors estimated the absorptivity and the heat capacity of carbon particles in the range 800–1200 K. Fecht and Johnson (1991) proposed a method for the accurate determination of the heat capacity of metastable undercooled liquid metals and alloys using electromagnetic levitation. Measurements on solid niobium samples in a levitation system TEMPUS (“tiegelfreies elektromagnetisches Prozessieren unter Schwerelosigkeit”) demonstrated the feasibility of this method (Wunderlich and Fecht, 1993; Wunderlich et al., 1993; Fecht and Wunderlich, 1994). A 10-mm solid niobium sphere was suspended in the center of a RF-heating coil in an ultrahigh-vacuum chamber (Wunderlich and Fecht, 1996). These preliminary measurements made it possible to determine the speciFc heat of metallic samples in space (Wunderlich et al., 1997; Egry, 2000). A sinusoidal modulation of the RF current in the heating coil was used to determine the heat capacity of metallic samples. 5.8. Modulated di5erential scanning calorimetry A new calorimetric technique was recently proposed, namely, modulated di0erential scanning calorimetry, MDSC (Gill et al., 1993; Reading et al., 1993; Boller et al., 1994). In MDSC, the sample’s temperature is sinusoidally varied about a constant ramp, so that T = T0 + Kt + A sin !t :
(5.18)
The resulting heating rate dT=dt thus oscillates about the underlying heating rate K. This approach allows one to separately measure the reversible heat Eow depending on the heat capacity of the sample, and the non-reversible heat Eow related to the latent heat of a Frst-order phase transition. Table 5.5 lists some works employing the MDSC technique.
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Table 5.5 Modulated di0erential scanning calorimetry (MDSC)a Item
Reference
Presentation of MDSC technique Compatibility of DSC and AC calorimetry Mathematical description of MDSC
Gill et al. (1993), Reading et al. (1993), Boller et al. (1994) Hatta (1994) Wunderlich et al. (1994), Wunderlich et al. (1996), Lacey et al. (1997) Hatta et al. (1995) Nishikawa and Saruyama (1995) Schawe (1995, 1996), Jones et al. (1997), Reading (1997) Hatta and Muramatsu (1996) Hensel et al. (1996), Ribeiro and Grolier (1999) Varma-Nair and Wunderlich (1996) Bohn et al. (1997) Baur and Wunderlich (1998), Merzlyakov and Schick (1999a, b) Boller et al. (1998)
NaNO2 MDSC employing light heating Interpretation of MDSC Experimental test of MDSC Polymers Chemical reactions Lead germanate, betaine arsenate, TGS Complex heat capacity Crystallization of indium BaTiO3 ; NaNO2 , 8OCB, Frst-order transitions SrTiO3 ; KH2 PO4 , PVA, DFTCE Pb nanoparticles embedded in Al matrix Pd 40 Ni10 Cu30 P20 , glass transition Crystallization of polymers
Hatta and Nakayama (1998) Kr[uger et al. (1998) Li et al. (1998) Hu et al. (1999) Dweck (2000), Schick et al. (2000)
a
TGS = triglycine sulfate, 8OCB = octyloxycyanobiphenyl, PVA = polyvinyl acetate, DFTCE = 1,2-diEuoro1,1,2,2-tetrachlorethane.
5.9. Commercial modulation calorimeter ACC-1 Sinku-Riko, Inc. has designed the Frst commercial modulation calorimeter ACC-1 (AC calorimeter). It employs modulated-light heating. With a liquid-nitrogen cryostat, the calorimeter operates in the 70–800 K range. The samples are of dimension 2×2×(0:01–0:3) mm3 for relative measurements and 4 × 4 × (0:1–0:3) mm3 for determinations of absolute values of the speciFc heat. The samples are placed in vacuum or a helium atmosphere. The modulation frequency is in the range 0:1–100 Hz. A digital lock-in ampliFer has been developed for infralow modulation frequencies. The calorimeter provides a programmable change of the mean temperature. The data are processed by a personal computer and displayed on a monitor or by means of a plotter. The inaccuracy of the measurements is 3% but the scatter of the data is about 0.01%. Determinations of the thermal di0usivity and the thermal conductivity of the samples also are possible. The calorimeter was commercialized by Sinku-Riko through the technical guidance of A. Ikushima and I. Hatta. The calorimeter ACC-1 was successfully employed in studies of high-temperature superconductors (Ishikawa et al., 1988; Kishi et al., 1988; Okazaki et al., 1990). Many authors studied phase transitions in solids (Kawaji et al., 1989; Gesi, 1992; Onodera et al., 1993; Irokawa et al., 1994; Gesi and Osaka, 1995; Haga et al., 1995a, b; Hatori et al., 1996; Tura et al., 1998). Tsuchiya (1991, 1993, 1995) has found structural changes in liquid Te; Ge15 Te85 and CdSb.
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Table 5.6 Calorimetric setups Item
Reference
Thin Flms, low temperatures Calorimeter with 3 He cryostat, 0.3–3 K High pressures, low temperatures Improvement of light-heating method Di0erential calorimeter, biological materials Automated calorimeter, 2–380 K Thin Flms, 0.4 –3:5 K Biological materials, 270 –400 K Photocalorimetric spectroscopy Automated high-resolution calorimeter SpeciFc-heat spectrometer, 0.01–3000 Hz Magnetic Felds up to 20 T Variable-depth calorimeter Microcalorimeter, low temperatures Bath modulation Magnetic Felds up to 8 T, 4.2–200 K SpeciFc-heat spectrometer, 10 –10 000 Hz Microcalorimeter, adsorbed gases Free-standing thin Flms Quench-condensed Flms, 0.1–7 K High-sensitivity calorimeter SpeciFc heat spectrometer, 0.01–10 000 Hz Calorimeter using modulation and relaxation modes Microcalorimeter, low temperatures Quench-condensed Flms, 0.4 –3 K Di0erential microcalorimeter, low temperatures Contactless calorimeter SpeciFc-heat spectrometer Multifrequency calorimeter, 0.5 –50 mHz Nanocalorimeter, low temperatures SpeciFc-heat spectrometer, 0.2–2000 Hz Microcalorimeter, 5 –300 K Microcalorimeter, 40 –160 K, high resolution Calorimeter with optical Fber light guide Modulated DSC setup Di0erential calorimeter Microcalorimeter for liquids Peltier microcalorimeter
Greene et al. (1972) Manuel et al. (1972) Eichler and Gey (1979) Ikeda and Ishikawa (1979) Dixon et al. (1982) Stokka and Fossheim (1982a) Suzuki et al. (1982) Imaizumi et al. (1983) Geraghty et al. (1984) Garland (1985) Birge and Nagel (1987) Schmiedesho0 et al. (1987) Wang and Campbell (1988) Chae and Bretz (1989) Graebner (1989) Calzona et al. (1990) Inada et al. (1990) Kenny and Richards (1990a, b) Geer et al. (1991b) Menges and von L[ohneysen (1991) Bednarz et al. (1992) Jung et al. (1992) Ema et al. (1993), Yao et al. (1998) Minakov and Ershov (1994) Birmingham et al. (1996) Carrington et al. (1996, 1997) Wunderlich and Fecht (1996) Birge et al. (1997) Ema and Yao (1997) Fominaya et al. (1997a) Korus et al. (1997) Marone and Payne (1997) Riou et al. (1997) GarFeld et al. (1998) Kr[uger et al. (1998) Maesono and Tye (1998) Yao et al. (1999) Moon et al. (2000)
Ogura et al. (1992) studied a polymer compound, vinylidene Euoride–triEuoroethylene. With a modiFed version of the calorimeter employing a liquid-helium cryostat, the temperature range can be extended down to 2 K (Castro and Burriel, 1995a, b). Hwang et al. (1992), Murayama et al. (1995), and PRerez et al. (1999) also performed measurements at liquid-helium temperatures. In concluding this section, a list of the modulation calorimetric setups described in detail is given in Table 5.6.
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Fig. 6.1. SimpliFed diagram of an interferometric dilatometer (Feder and Charbnau, 1966).
6. Modulation dilatometry 6.1. Methods of dilatometry, a brief review At present, methods for measuring the dilatation of solids provide a sensitivity of the order of 10−10 m and even better. However, diIculties of high-temperature dilatometry are caused by the poor stability of the samples rather than by the lack of sensitivity. This is why one had to accept data on thermal expansivity averaged within wide temperature intervals. Important improvements in this Feld have been made in the last decades. Along with a signiFcant progress in traditional dilatometry, two new techniques have appeared, modulation dilatometry and the dynamic technique. Most sensitive dilatometers employ optical interferometers and capacitance sensors. Interferometric measurements became much easier by using lasers, owing to the high temporal and spatial coherence of the laser beam. 6.1.1. Optical methods Feder and Charbnau (1966) have designed a sensitive laser dilatometer. Changes in the sample’s length were measured by a Fizeau-type interferometer (Fig. 6.1). The light beam is reEected from the bottom optical Eat upon which the sample rests and from the top Eat. The reEected beams produce visible interference fringes. The fringe shifts thus depend only upon the changes in the sample’s length. A He–Ne laser serves as the light source, and a special system was designed for automatic fringe counting. The assembly, consisting of a prism, two adjustable slits, and two phototubes, is Frst rotated, so that the fringes are perpendicular to the apex of the prism. Each fringe is then split into two parts with each phototube seeing one-half of the fringes. By connecting the phototubes to a two-pen chart recorder, the fringe shifts are recorded as a series of sine waves. This double-recording system senses changes in the direction of the fringe motion. Temperature control is achieved by immersing the tube containing the sample in a regulated oil bath. The accuracy of the measurements depends only on the precision with which the wavelength of the light is known and the shift in fringe pattern is determined.
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Fig. 6.2. Interferometric dilatometer designed by Jacobs et al. (1970). L—lens, M—modulator, P—polarizer, SF—spatial Flter, PM—photomultiplier.
Jacobs et al. (1970) have developed a very sensitive dilatometer based on the dependence of Fabry–Perot resonances on the mirror separation. The sample governs a space between the mirrors (Fig. 6.2). The light source consists of variable frequency sidebands obtained by modulation of a stabilized He–Ne laser. A change in the sample’s temperature causes a change in the resonance frequencies. The modulation frequency is adjusted until one of the sidebands coincides with a Fabry–Perot resonance. When the temperature changes, a new modulation frequency is found for the maximum transmittance. The change in this frequency shows the relative change in the sample’s length. The sample has the form of a hollow cylinder 10 cm long. The laser beam collimated by a lens passes through a modulator. A polarizer and a quarter-wave plate serve as an isolator against reEections back. The polarizer also suppresses the carrier frequency, so that only the sidebands are transmitted. Broadband tuning in the range 10–480 MHz is accomplished by means of an oscillator and ampliFer. The precision of the dilatometer is limited only by the laser instability, which is of the order of 10−9 . This technique is a good Fnding for traditional dilatometry. Suska and Tschirnich (1999) designed a sensitive dilatometer based on a Fizeau-type interferometer with two corner-cube reEectors. In measurements on 1-m bars, the overall uncertainty of the measurements is smaller than 2 × 10−8 K −1 . 6.1.2. Capacitance dilatometers Johansen et al. (1986) designed a sensitive capacitance dilatometer. The dilatometer integrated in an oven is of the parallel-plate capacitor type. A Peltier element provides Fne regulation of ◦ ◦ the oven. In the range −60 C to 150 C, the oven can be stabilized to better than 10−4 K. For large dilatations, a HP 4192 low-frequency impedance analyzer measures the capacitance. For a capacitance near 15 pF, the sensitivity obtained is 10−3 pF. A computer repeatedly reads and averages the analyzer’s data. For the submicron range, the detection system is based on a manual capacitance bridge and a lock-in ampliFer. The ampliFer’s output signal is calibrated versus the deviation in capacitance between a reference and the dilatometer’s capacitor. In this case, the sensitivity is 7 × 10−12 m. A quartz-crystal thermometer measures the temperature of the dilatometric cell. Johansen (1987) modiFed this dilatometer to operate in a modulation regime. Rotter et al. (1998) developed a miniature capacitance dilatometer suitable for measuring the thermal expansion and magnetostriction of small samples and samples of irregular shape.
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Fig. 6.3. Principle of dynamic dilatometry developed by Miiller and Cezairliyan (1982). This technique was successfully employed in studies of thermal expansion of refractory metals (Miiller and Cezairliyan, 1982, 1985, 1988, 1990, 1991).
The active length of the sample may be smaller than 1 mm. An AC bridge measures the capacitance. The absolute resolution is about 10−10 m. The dilatometer was tested in the range 0:3–200 K and in magnetic Felds up to 15 T. 6.1.3. Dynamic techniques Miiller and Cezairliyan (1982) developed a dynamic technique involving resistively heating the sample from room temperatures to above 1500 K in less than 1 s. The sample is mounted in a chamber providing measurements either in vacuum or in a gas atmosphere (Fig. 6.3). A photoelectric pyrometer capable of 1200 evaluations per second determines its temperature. Simultaneously, a polarized-beam laser interferometer measures the expansion of the sample. The sample has the form of a tube with parallel optical Eats on opposite sides. The distance between the Eats, 6 mm, represents the length of the sample. The sample acts as a double reEector in the path of the light beam. The interferometer is thus insensitive to translational motion of the sample. The rotational stability is monitored by reEecting the beam of an auxiliary laser from a third optical Eat on the sample. A small rectangular hole in the wall of the sample, 0:5 × 1 mm2 , serves as a blackbody model for the temperature measurements. During the pulse heating, a dual-beam oscilloscope displays the traces of the radiance from the sample and of the corresponding shift in the fringe pattern. This system has two very important advantages: (i) the measurements relate to the blackbody temperature, and (ii) only a central portion of the sample is involved in the measurements, so that there is no need to take into account the temperature distribution along the sample. By using this technique, the authors investigated the thermal expansion of tantalum, molybdenum, niobium, and tungsten at high temperatures (Miiller and Cezairliyan, 1982, 1985, 1988, 1990, 1991). Righini et al. (1986b) designed a dilatometer that correlates the thermal expansion of the sample to its temperature proFle. An interferometer measures the longitudinal expansion of the sample, while a scanning optical pyrometer determines its temperature proFle (Fig. 6.4). A typical spacing between two consecutive measurements is 0:35 mm with a pyrometer viewing area 0:8 mm in diameter. Two massive brass clamps maintain the ends of the sample close to room temperature and provide steep temperature gradients towards the ends. Two thermocouples spot-welded at the ends of the sample measure its temperature in the regions where pyrometric
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Fig. 6.4. SimpliFed diagram of a dynamic dilatometer (Righini et al., 1986b).
measurements are impossible. A corner cube retroreEector is attached to the lower (moving) clamp, while the beam bender is attached to the upper clamp. The resolution of the interferometer is about 0:15 m, and 2000 measurements per second are feasible. All the results proceed to a data-acquisition system. Radiance temperatures measured by the pyrometer are transformed into true temperatures by either the resistivity of the sample or data on the normal spectral emittance. The measurements on a niobium sample (Righini et al., 1986a) lasted from 0:3 s (fast) to 2:2 s (slow). For the proFle measurements, fast experiments are preferable because the portion of the proFle not known from the scanning pyrometer is below 6%. For slow experiments, this Fgure increases to 20%. On the other hand, in fast measurements the thermal-expansion polynomial is deFned by few data points limited by the speed of rotation of the mirror. A compromise must be found between a better knowledge of the temperature proFle and a better deFnition of the thermal-expansion polynomial. The authors stressed that an ideal experiment with this technique would be either to bring the entire sample to the high temperature or to limit the measurements to a central portion of the sample. 6.2. Principle of modulation dilatometry, wire samples Measurements of the ‘true’ expansivity, the thermal expansion coeIcient within a narrow temperature interval, are possible using a modulation technique. It involves oscillating the sample’s temperature about a mean value and measuring corresponding oscillations in the sample’s length. Under such conditions, the linear expansivity is measured directly. This approach is very attractive because it ignores irregular changes in the sample’s length due to external disturbances. In modulation dilatometry, only those changes in the sample’s length are measured that reversibly follow the temperature oscillations. This seems the best way to get around the main problems peculiar to the traditional dilatometry at high temperatures: one measures only what is necessary, while all unwanted disturbances appear beyond the measurements. The measurements are possible with temperature oscillations of the order of 0:1 K and less. At present, even a resolution of 1 K at high temperatures is a great improvement over the traditional
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Fig. 6.5. Modulation measurements of thermal expansivity. Only oscillations of the length are measured whose frequency equals the frequency of the temperature oscillations in the sample (Kraftmakher and Cheremisina, 1965).
methods. To measure the oscillations in the sample’s length, one can employ the most sensitive techniques. Modulation dilatometry was proposed long ago (Kraftmakher and Cheremisina, 1965). It was applied to the study of the thermal expansion of metals and alloys at high temperatures (Kraftmakher, 1967b, 1972; Glazkov, 1985, 1987; Glazkov and Kraftmakher, 1986). This technique was reviewed in several papers (Kraftmakher, 1973b, 1978a, 1984, 1989). However, it gained no recognition. After 20 years, it has been reinvented and applied to nonconducting materials by Johansen (1987). In the Frst modulation measurements, an AC current or a DC current with a small AC component heated a wire sample (Fig. 6.5). The upper end of the sample is Fxed, while the lower one is pulled by a load or a spring and is projected onto the entrance slit of a photomultiplier. The AC voltage at the photomultiplier’s output is proportional to the amplitude of the oscillations in the sample’s length. The temperature oscillations are determined from either the electrical resistance of the sample or the radiation from it. They are also available from the speciFc heat of the sample. When a DC current I0 with a small AC component heats the sample, the linear expansivity # = (1=l) dl=dT obeys the expression # = mc!V=2lKI0 U :
(6.1)
Here m; c, and l are the mass, speciFc heat, and length of the sample, U is the AC voltage across it, V is the AC component at the output of the photomultiplier, and K is the sensitivity of the photomultiplier to the elongation of the sample. The sensitivity is available from static measurements: K = dV0 =dl, where V0 is the DC voltage at the photomultiplier’s output. A circuit can be assembled whose balance is independent of the AC component of the heating current. In this case, the AC signal at the photomultiplier’s output is balanced by a variable mutual inductance M with the heating current passing through its primary winding. The voltage at the secondary winding becomes equal to the AC voltage at the output of the photomultiplier when # = mc!2 M=2I0 lR0 K :
(6.2)
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Fig. 6.6. Direct comparison of thermal expansivities of two samples. The oscillations in the length of the sample under study, AB, and of the reference sample, BC, balance each other (Kraftmakher, 1967d).
The measuring system incorporates a selective ampliFer tuned to the modulation frequency. The dilatometer thus becomes insensitive to irregular mechanical perturbations or to oscillations of other frequencies. When an AC current heats the sample, the expansivity is given by # = 2mc!V=lPK ;
(6.3)
where P is the mean electric power supplied to the sample. The modulation frequency, in the range 10–100 Hz, is assumed to satisfy the criterion of adiabaticity. Otherwise, the temperature oscillations obey formulas for a nonadiabatic regime. 6.3. Di5erential method In the di0erential method (Kraftmakher, 1967d), a wire sample consists of two portions joined together: a sample under study and a reference sample of known linear expansivity (Fig. 6.6). The two portions are heated by DC currents from separate sources and by AC currents from a common oscillator. The temperature oscillations in the two portions are of opposite phase. By adjusting the AC components of the heating currents, the oscillations in the length of the reference portion completely balance those of the portion under study. Now the photomultiplier acts only as a null indicator and any variations in its sensitivity, as well as in the intensity of the light source, etc., do not contribute. To calculate the expansivity, one has to determine the temperature oscillations in the two portions. If the corresponding speciFc heats are known, a simple relation holds when the oscillations balance each other: #1 I01 U1 l1 =m1 c1 = #2 I02 U2 l2 =m2 c2 :
(6.4)
Subscripts 1 and 2 refer to the main and the reference portions of the sample, respectively. The reference sample is kept at a constant mean temperature, and all the related quantities are
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Fig. 6.7. Linear thermal expansivity of platinum and tungsten measured by using modulation dilatometry (Kraftmakher, 1967b, 1972). The nonlinear increase in the expansivity was attributed to vacancy formation.
constant. Hence, # = BU2 c1 =I01 U1 ;
(6.5)
where B is a coeIcient of proportionality. The measurements are thus reduced to nullifying the oscillations in the length of the composite sample and measuring the DC current in the main portion and the AC voltages across the two portions. This technique was employed in studies of the thermal expansion of platinum (1000–2000 K) and tungsten (2000–2900 K). The samples were 0:05 mm thick. Their mean temperature was determined from the electrical resistance, while the temperature oscillations, of about 1 K, were calculated from the speciFc heat. Similar wires kept at constant mean temperatures served as reference samples. The sensitivity of the setup was about 1 nm. In both cases, a nonlinear increase of the expansivity is clearly seen (Fig. 6.7). It was attributed to vacancy formation in the crystal lattice (Kraftmakher, 1967b, 1972). 6.4. Bulk samples Another version of modulation dilatometry is suitable for comparatively bulk samples, such as rods and strips (Fig. 6.8). To eliminate the cold-end e0ects, the temperature oscillations occur
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Fig. 6.8. Modulation dilatometer for bulk samples (Kraftmakher and Nezhentsev, 1971). ET—electromechanical transducer, PD—photodiode.
only in a central portion of the sample. A mains current heats the sample. A small current of the same frequency, but with its phase linearly varying with time, is added to the mains current in the central portion. The superposition of the two currents causes oscillations in the power dissipated in the central portion, and its temperature oscillates around a mean value. The modulation period is in the range 1–10 s. A thermocouple measures the mean temperature and the temperature oscillations in the central portion of the sample. An electromechanical transducer, a small earphone, serves for the compensation. It is attached to the sample. A blade glued to the earphone’s membrane is illuminated and its image is projected onto a photodetector. The output voltage of the photodetector is partly compensated and then fed to an ampliFer. The transducer is connected to the ampliFer’s output to balance the changes in the sample’s length. Owing to the high gain of the ampliFer, the system provides almost complete compensation, while the sensitivity is about 10 nm. The changes of the current in the transducer are proportional to the changes in the sample’s length. A recorder records these oscillations, along with the temperature oscillations in the sample (Kraftmakher and Nezhentsev, 1971). A thin wire heated by an electric current also can serve as a compensator. One of its ends is attached to the sample and the other is pulled by a spring. A blade mounted at the upper end of the spring is projected onto a photodiode. Since the temperature oscillations in the sample are small, only small changes in the temperature of the wire are necessary for compensation. The mean temperature of the wire is constant, and the changes in its length are proportional to the changes of the heating current. The time response and linearity of a 50-m tungsten wire at 1500 K appeared to be suIcient. To determine absolute values of the thermal expansivity, the compensator requires a calibration. Samples of known thermal expansivity are also usable. In many cases, it is enough to know the temperature dependence of the expansivity, so that relative values are suIcient. Using mean values of the expansivity in wide temperature ranges available from traditional measurements can normalize the data. To verify this technique, measurements on a nickel sample were performed in the range 700–1400 K. The length of the sample was 200 mm, while the temperature oscillations were created in its central portion 40 mm long. The modulation period was 10 s, and the temperature oscillations amount to about 5 K. A thin tungsten wire served as the compensator.
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Fig. 6.9. Interferometric modulation dilatometer (Glazkov and Kraftmakher, 1983). Temperature oscillations are created only in the central portion of the sample. A piezoelectric transducer PT balances oscillations in the sample’s length.
6.5. Interferometric modulation dilatometer Oscillations in the sample’s length are measurable by various methods, including those of highest sensitivity. The interferometric method is one such technique. In the interferometric modulation dilatometer (Glazkov and Kraftmakher, 1983), samples are in the form of a thick wire or a rod (Fig. 6.9). The upper end of the sample is Fxed, while a small mirror M 1 is attached to its lower end. A DC current from a stabilized source heats the sample. A small AC current is fed to a central portion of the sample through thin wires. The temperature oscillations thus occur only in this portion. To prevent an o0shoot of the AC current to the upper and lower portions of the sample, a coil of a high AC resistance is connected in series with the sample (not shown in the Fgure). The beam of a He–Ne laser passes through a beam splitter and falls onto the mirror M 1 and a second mirror M 2 attached to a piezoelectric transducer. A photodiode senses the intensity of the interference pattern. An ampliFer ampliFes its output voltage. The ampliFed voltage is applied to the transducer with such a polarity that the oscillations of both mirrors are in phase. Owing to the high gain of the ampliFer, the displacements of the mirror M 2 follow those of the mirror M 1. The AC voltage applied to the transducer is therefore strictly proportional to the oscillations in the sample’s length. A selective ampliFer tuned to the modulation frequency ampliFes this voltage, and a lock-in detector measures it. The reference voltage for the detector is taken from the oscillator supplying the AC current to the central portion of the sample. The output voltage of the detector is proportional to the amplitude of the oscillations in the sample’s length. A 0.3-mm platinum wire was used to check the dilatometer. The length of the sample was 150 mm, while the temperature oscillations were generated in its central portion 40 mm long. The data have shown good linearity and the possibility of performing measurements with temperature oscillations of the order of 0:1 K. The results were almost independent of the intensity of the laser beam. The wires conFning the central portion were 50 m in diameter. Clearly, they disturb the temperature around the points where they are welded to the sample. Adjusting
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Fig. 6.10. Modulation dilatometer for nonconducting samples, schematically (Johansen, 1987).
the current passing through the wires reduces this e0ect. Using an additional current to heat the wires, the mean temperature of the wires can be made close to that of the sample. A somewhat more diIcult task is to create equal temperature oscillations in the sample and the wires. A micropyrometer and a photodiode were used to compare the mean temperatures and the temperature oscillations along the sample. In the central portion, variations of the mean temperature, 1500 K, did not exceed 2 K when an additional heating of the wires was employed. Without such a heating, the decrease of the temperature at the points of welding the wires amounted to 30 K. The amplitude of the temperature oscillations inside the central portion was constant within 1%, while no temperature oscillations were found beyond it. Radial temperature di0erences in the sample are negligible because of its small thickness and high thermal conductivity. With the radiation heat exchange, the di0erences do not exceed 0:2 K at 2000 K and decrease with decreasing the temperature. The relations between the amplitudes and the phases of the temperature oscillations inside the sample and on its surface are also favorable. To check the validity of a modulation dilatometer, it is probably suIcient to verify that the results obtained do not depend on the modulation frequency. The uncertainty in the oscillations of the sample’s length is about 1%. The errors in evaluating the linear expansivity depend mainly on the uncertainty in measuring the temperature oscillations. 6.6. Nonconducting materials Johansen (1987) employed a computerized capacitance dilatometer (Johansen et al., 1986) in a regime of periodic changes of the sample’s temperature. A Peltier module provides AC heating of a copper block containing the sample (Fig. 6.10). Capacitance measurements are made with a precision bridge and a lock-in ampliFer as the imbalance detector. A 10−11 m resolution is achieved over a large dynamic range, of about 10−6 m. Unlike Joule heating, the Peltier e0ect introduces no DC temperature increment. The temperature of the copper block ◦ could be controlled at a Fxed temperature point or in a linear sweep mode, in the range −60 C ◦ to 150 C. To check the dilatometer, the linear thermal expansivity of Rochelle salt, NaKC4 H2 O6 · 4H2 O; ◦ was measured near 24 C. The salt undergoes an orthorhombic-to-monoclinic phase transition at this temperature. The modulation frequency was 2:5 mHz, and the amplitude of the temperature oscillations was about 0:02 K.
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Fig. 6.11. Measurement of small periodic displacements using Fabry–Perot interferometer (Bruins et al., 1975).
6.7. Measurement of extremely small periodic displacements Several methods exist to measure very small periodic displacements. The periodic nature of the displacements allows one to employ selective ampliFers and lock-in detectors. This enhances the sensitivity of the measurements far above the sensitivity to usual (not periodic) displacements. The most convenient methods to measure periodic displacements employ laser interferometry. Bruins et al. (1975) developed one such technique (Fig. 6.11). A spherical Fabry–Perot interferometer was used to determine small piezoelectric constants, but the authors pointed out that the method could be adapted to other measurements. The interferometer has several advantages over other interferometric arrangements. First, it is inherently much more sensitive than the Michelson or other two-beam interferometer, because of the high sharpness of the multiple-beam interference fringes. Second, it is a compact instrument suitable for a temperature-controlled environment. A piezoelectric transducer served as a reference. The vibrations of about 4 × 10−14 m were measured with 5% accuracy. The theoretical sensitivity in a well-isolated environment was estimated to be 10−15 m. The method is also usable as a null technique. A voltage of the same frequency should be applied to the transducer and its amplitude and phase adjusted to exactly balance the displacements of the sample. The sensitivity is suIcient for any modulation measurements of thermal expansion. However, the authors did not consider this case. Fanton and Kino (1987) designed a simple setup for measurements of small periodic displacements (Fig. 6.12). The laser beam passes through a polarizing beam splitter and a birefringent ◦ Wollaston prism with its optical axis oriented at 45 to the direction of polarization. The prism ◦ splits the beam into equal-amplitude beams angularly separated by 0:5 and focused to two spots on the sample. The reEected beams recombine in the Wollaston prism, pass through the splitter, and interfere on a photodetector. The beam focused onto an undisturbed portion of the sample serves as a reference. At the focus of the other beam, the sample is heated periodically. The resulting thermal expansion of the sample causes phase modulation of the beam, which is converted to intensity modulation when the two beams interfere. Both beams pass through the same optical components making the system resistant to vibrations and drift. A lock-in ampliFer reduces the inEuence of noise and disturbances of other frequencies. The sensitivity of the system was checked by measuring vibrations of a piezoelectric transducer split into halves, one stationary, and one active. The background optical noise of the system was 3:4 × 10−14 m.
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Fig. 6.12. Setup for measuring small periodic displacements (Fanton and Kino, 1987). PD—photodetector.
The system was also used to obtain a thermal image of a Eawed conducting strip, a 25-nm nickel Flm on a silicon wafer. The strip was notched and then heated by passing an 8-kHz current through it. The actual temperature oscillations in the strip were roughly 10 mK. The authors pointed out that it is possible to modulate the laser at a frequency close to that of the vibrations and to obtain a signal of the di0erence frequency. The sensitivity achievable by the above techniques is much better than is necessary for modulation dilatometry. Thermal expansivity becomes measurable even on very short samples. In particular, it can be determined along the sample’s thickness, so that the problem of temperature gradients in the sample would be completely avoided. 7. Modulation measurements of electrical resistance, thermopower, and spectral absorptance 7.1. Temperature derivative of resistance The modulation technique allows one to directly measure the temperature derivative of electrical resistance. The method consists in oscillating the sample temperature around a mean value and measuring the oscillations in the sample’s resistance along with the temperature oscillations (Kraftmakher, 1967a). A nickel sample was heated by a DC current from a stabilized source and by a modulated AC current. A thermocouple measured the temperature oscillations in the sample. Simultaneously, measurements were made of the voltage oscillations that appear due to the DC current passing through the oscillating sample’s resistance. To eliminate the cold-end e0ects, the measurements were carried out on a central portion of the sample. A two-channel recorder recorded the AC components of the thermocouple’s voltage and of the voltage drop across the central portion of the sample. Data on the temperature derivative of resistance clariFed the nature of the anomaly at the Curie point. The method revealed the point defect contributions to the electrical resistivity of aluminum and platinum (Fig. 7.1). Salamon et al. (1969) used a similar method in studies of the resistivity of chromium near its NReel point, by means of the modulated-light heating. Lederman et al. (1974) described this technique in more detail. The speciFc heat and the temperature derivative of resistance of iron near its Curie point were measured simultaneously (Fig. 7.2). The temperature within the
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Fig. 7.1. Temperature coeIcient of resistance (TCR) of aluminum and platinum. The nonlinear increase was attributed to vacancy formation (Kraftmakher and Sushakova, 1972, 1974).
Fig. 7.2. Measurement of temperature derivative of resistance by using modulated-light heating. Oscillations in the temperature and in the resistance of the sample are recorded simultaneously (Lederman et al., 1974).
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Fig. 7.3. Measurement of thermopower by using modulated-light heating (Hellenthal and Ostholt, 1970).
furnace was linearly swept at a rate of 1 K min−1 or less. The modulation frequency was 8 Hz. Lock-in ampliFers measured the AC voltages corresponding to the temperature and resistance oscillations. Chaussy et al. (1992) reinvented this technique and employed it to determine the anomaly in the resistivity of erbium near its NReel point (Terki et al., 1992). Modulation measurements of the temperature derivative of resistance were carried out on CuZn (Simons and Salamon, 1971), V3 Si (Chu and Testardi, 1974), iron (Shacklette, 1974; Kraftmakher and Pinegina, 1974), solid electrolytes (Vargas et al., 1976, 1977), samarium (Kraftmakher and Pinegina, 1978), PtRh, nickel, and rhodium (Glazkov, 1985, 1987, 1988). 7.2. Direct measurement of thermopower Among all the modulation techniques, measurements of thermopower (the Seebeck coeIcient) are the simplest. The method consists in measuring the same periodic temperature oscillations by the thermocouple under study and a reference one. Three groups have independently proposed this technique (Freeman and Bass, 1970; Hellenthal and Ostholt, 1970; Kraftmakher and Pinegina, 1970). It was reinvented several times (Korn and M[urer, 1977; Kawai et al., 1978; Papp, 1984; Howson et al., 1989, 1990). Freeman and Bass (1970) and Hellenthal and Ostholt (1970) described very similar methods to measure the Seebeck coeIcient. A modulated light heats the thermocouple junction (Fig. 7.3). A lock-in ampliFer measures the signal from the thermocouples. It is possible to compensate the signal at the ampliFer’s input or to continuously record it. The high sensitivity of the method allows one to study subtle e0ects, e.g., changes in the thermopower resulting from a deformation (Hellenthal and Ostholt, 1970). In measurements at high temperatures, a small low-inertia furnace created temperature oscillations measured by two thermocouples (Kraftmakher and Pinegina, 1970; Kraftmakher, 1971a). Many other workers carried out modulation measurements of thermopower (Chaikin and Kwak, 1975; Kettler et al., 1982, 1984, 1986; Kawai et al., 1984; Yu et al., 1988; Mangelschots et al., 1992; Oussena et al., 1992; Xu et al., 1992; Lin et al., 1993; Resel et al., 1996; Goto et al., 1997; Kato et al., 1999).
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7.3. Spectral absorptance The radiation properties of materials are emittance, absorptance, reEectance, and transmittance. They control the rate of heat transfer by radiation between noncontacting bodies, and between a body and its surroundings (for a review see Richmond, 1984). Spectral emittance and absorptance are equal only under thermodynamic equilibrium. No reliable data on these characteristics are available for many important materials. The number of such materials grows rapidly, and the data are necessary in wide ranges of temperatures and wavelengths. Spectral absorptance is measurable even at low temperatures when thermal radiation is still negligible. The absorptance may be related to total or partial radiation from thermal sources of various temperatures or from any other source of radiation. In particular, absorptance of solar radiation is very important for many applications. For samples of very high reEectance, measurements of spectral absorptance provide an advantage. For instance, it is very diIcult to distinguish between samples with reEectances 0.98 and 0.99. In contrast, the di0erence between absorptances 0.01 and 0.02 is easy to measure. Two calorimetric techniques are known for the determination of spectral absorptance. In steady-state measurements, the power of the absorbed radiation is calculated from the increment in the temperature of the sample and the heat transfer coeIcient (Biondi, 1954, 1956). With a dynamic approach, the data are obtainable from the heating rate and the heat capacity of the sample (Annino et al., 1984; Szil et al., 1985; MogyorRosi et al., 1986). In the modulation measurements, the surface of the sample is exposed to periodic pulses of radiation. The absorption of the radiation leads to an increase in the mean temperature of the sample and to periodic temperature oscillations. For short modulation periods, the amplitude of the temperature oscillations is proportional to the period and to the absorbed power and inversely proportional to the heat capacity of the sample. The basic relation of the modulation calorimetry may be used in a straightforward manner to measure the absorbed radiation. The proposed approach involves a compensation technique that requires no data on the heat capacity of the sample (Kraftmakher and Tarasenko, 1987). Pulses of electric current are passed through the sample, being supplementary to the pulses of the absorbed radiation (Fig. 7.4). When the power of the electric pulses is equal to the power of the absorbed radiation, the total power applied to the sample becomes constant, and the temperature oscillations in the sample vanish. A device with a perfect blackbody measures the power of the incident radiation. The advantages of this approach are evident: (i) high sensitivity because very small temperature oscillations are measurable, (ii) wide temperature range, and (iii) enhancement of the accuracy by the compensation. Restrictions also exist because thin conducting samples are necessary for the measurements. A special setup was assembled to check the method (Fig. 7.5). A 500-W halogen lamp acts as the source of radiation. A monochromator provides monochromatic radiation to irradiate the sample. An electrical relay driven by a pulse generator serves to chop the radiation and to switch on and o0 the current passing through the sample. The samples, metal strips 40 × 2 × (0:05– 0:1) mm3 in size, are placed in a vacuum chamber. Thin wires conFne their central portion, 20 mm long. A thermocouple junction is pasted to the backside of the sample, without an electric contact. The signal of the thermocouple is ampliFed and recorded. The modulation period is 10 s and the temperature oscillations caused by the radiation range from 0.01 to 0:1 K.
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Fig. 7.4. Principle of measuring spectral absorptance by employing compensation method (Kraftmakher and Tarasenko, 1987). Fig. 7.5. Apparatus for measuring spectral absorptance (Kraftmakher and Tarasenko, 1987).
The measurements were performed using two methods. First, the amplitude of the electric pulses was increased gradually causing a decrease of the temperature oscillations in the sample. When the electric power becomes larger than the power of the absorbed radiation, the phase ◦ of the temperature oscillations changes by 180 . A plot of the amplitude of the temperature oscillations versus the electric power supplied to the sample allows one to determine the absorbed power. The power of the incident radiation was measured using a black painted sample, for which the spectral absorptance was considered unity. Second, one surface of the strip under study was painted black. The temperature oscillations in it were measured when the radiation fell, in turn, on both surfaces. The ratio of the amplitudes of the temperature oscillations equals the spectral absorptance. When the absorbed power is greater than 1 mW cm−2 , the scatter of the data is less than 5%. To enhance the sensitivity and extend the wavelength range, interference Flters should be used instead of a monochromator. They provide a higher radiation power necessary for samples of low absorptance. At higher temperatures, the temperature oscillations were detected through the resistance of the samples. A DC current heated the samples, while a modulated AC current created pulses of additional electric power. The compensation technique has certain advantages over traditional methods of measuring spectral absorptance. The measurements are possible on strips of conducting materials (metals, alloys, semiconductors), thin conducting or insulating coatings on such strips, or conducting coatings on insulating substrates of any thickness. 8. Noise thermometry of wire samples 8.1. Principles of noise thermometry Thin wire samples heated by an electric current passing through them are well suited for modulation measurements at high temperatures. However, an accurate determination of their temperature poses a serious problem. It is solvable by measurements of the thermal noise of the samples. As was theoretically shown by Nyquist, the mean squared thermal-noise voltage
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Fig. 8.1. Block diagram of a correlation ampliFer. Fig. 8.2. Correlation ampliFer with compensation. Anticorrelated voltages, Ua and −Ua , nullify output signal of the multiplier regardless of the gain of the channels (Kraftmakher and Cherevko, 1972a, b).
generated by a resistor in a narrow frequency band Tf equals TU 2 = 4kB RT Tf ;
(8.1)
where R is the resistance, kB is Boltzmann’s constant, and T is the absolute temperature of the resistor. After ampliFcation by means of a wide-band ampliFer, the noise voltage becomes ∞ 2 U = 4kB RT K 2 df ; (8.2) 0
where K is the ampliFer’s gain depending on frequency. Noise thermometers are in use since the pioneering work by Garrison and Lawson (1949). However, thermal noise of low-resistance samples may appear to be comparable with the inherent noise of an ampliFer. The correlation method of measurements is useful in such cases, and it was already employed in noise thermometers (e.g., Shore and Williamson, 1966; Storm, 1970). 8.2. Noise correlation thermometer In a correlation device, an electric signal U (e.g., thermal noise of a resistor) is fed to the inputs of two similar ampliFers (Fig. 8.1). Their output signals contain the ampliFed input voltage, U , and inherent noise voltages, U1 and U2 . The signals proceed to a multiplier and then to an integrating circuit. The result of the multiplication is V = K1 K2 (U + U1 )(U + U2 ) = K1 K2 (U 2 + UU1 + UU2 + U1 U2 ) :
(8.3)
Here K1 and K2 are total gains of the two channels. Since the inherent noise voltages of the ampliFers, U1 and U2 , are uncorrelated with each other and with the common input signal U , the corresponding products vanish after averaging. The mean output voltage is thus proportional to the square of the common input signal. However, it is sensitive to the gains of the channels: V = K1 K2 U 2 :
(8.4)
The proposed correlation ampliFer (Kraftmakher and Cherevko, 1972a) di0ers from those previously described in that it employs a compensation method (Fig. 8.2). The noise voltage
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Fig. 8.3. Noise correlation thermometer operating under periodic changes of sample’s temperature (Kraftmakher and Cherevko, 1974).
to be measured, Uc (correlated), is fed to the two channels of the ampliFer, and the ampliFed voltages proceed to a multiplier. For the compensation, two anticorrelated voltages, Ua and −Ua , are also fed to the channels of the ampliFer. They are taken from an independent noise generator and a phase splitter. Now the signal after averaging is V = K1 K2 (Uc2 − Ua2 ) :
(8.5)
When the RMS value of the compensating voltage equals that of the voltage to be measured, the Fnal signal is zero, regardless of the gains of the channels. The correlation ampliFer with compensation was employed as a noise thermometer for measurements on a 50-m platinum wire heated electrically in vacuum (Kraftmakher and Cherevko, 1972b). To simplify the calculations, two identical samples were used, with resistances of about 10 \ at 273 K. They were connected in series relative to a DC source and in parallel relative to the inputs of the correlation ampliFer. At various DC currents, the thermal noise was measured in a 60 –150 kHz frequency band along with the resistance of the samples. In the range 1200 –2000 K, the results have shown good agreement between the temperature derived from the noise measurements and that calculated from the resistivity of platinum taking into account the point-defect contribution. The scatter of the experimental points was about 1%, while the systematic deviation was several times smaller. 8.3. Determination of the temperature derivative of the resistance The noise correlation thermometer with compensation is also eIcient in a regime when the sample’s temperature periodically changes (Kraftmakher and Cherevko, 1974). The changes in the thermal noise and in the electrical resistance are measured to calculate the temperature derivative of the resistance and the heat transfer coeIcient of the sample. A DC current heating the sample oscillates with a period suIciently long to achieve thermal equilibrium (Fig. 8.3). The changes in the resistance of the samples corresponding to certain changes in the heating current were determined beforehand by a bridge circuit. This was suIcient to calculate the changes in the resistance corresponding to the temperature changes. It is easy to see that the temperature derivative of the resistance obeys the relation R = R1 =[T(RT )=TR − T2 ] = R2 =[T(RT )=TR − T1 ] :
(8.6)
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Here R1 and R2 are the resistances of the sample at the initial, T1 , and Fnal, T2 , temperatures, T(RT ) is a quantity available from the measured changes in the thermal noise, and TR is the corresponding change in the resistance. The heat transfer coeIcient of the sample is obtainable in a similar way: P = R1 TP=[T(RT ) − T2 TR] = R2 TP=[T(RT ) − T1 TR] ;
(8.7)
where TP is the increment in the heating power necessary to change the sample’s temperature from T1 to T2 . The measurements were performed on 8-m tungsten wires placed in an evacuated and sealed-o0 glass bulb. The resistance of the two samples connected in parallel was about 11 \ at 273 K. The modulation period was 120 s, and the amplitude of the temperature changes about 50 K. The system was calibrated using resistors at room temperature connected to the inputs of the correlation ampliFer in series with the samples. The quantity T(RT ) was calculated as an average from 10 to 20 periods of the temperature variations. The current density necessary to heat up a wire sample of diameter d to a certain temperature is proportional to d−1=2 . Measurements on thin samples allow one to check the Nyquist formula under nonequilibrium conditions when the current density is of the order of 108 A m−2 . The results have conFrmed the validity of the Nyquist formula under such conditions. A more diIcult task is to apply noise measurements to direct determinations of small temperature oscillations in a sample.
9. Electronic instrumentation for modulation measurements Modulation techniques require numerous electronic instruments. However, the necessary equipment is now readily available. SpeciFc requirements for this equipment are quite moderate. A list of electronic instruments and their features important for modulation measurements is given below. This equipment can be used to assemble modulation setups and to further develop the modulation techniques. DC sources are necessary for heating the samples to a desired mean temperature. Their main features are upper limits of the voltage and current, smallness of pulsations in the output voltage, stability, and convenience of regulation. Modern stabilized sources are very suitable, and any device with corresponding limits of the voltage and current is usable. Low-frequency oscillators provide modulation of the heating power. The main requirements for them are the stability of the frequency and amplitude of the output voltage, small distortions, a suIcient output power, and compatibility with a load, a sample or an electric heater. The maximum power is achieved when the internal resistance of the source equals the load resistance. Low-frequency power ampli@ers supply high currents to heat thick samples. They must provide a suIcient output power, stability, small distortions, and be compatible with a load. Light sources are needed for modulated-light heating. They must provide the necessary radiation power and ensure its stability. Usually, incandescent lamps and lasers are employed. Meters of electric current and voltage determine oscillations of the heating power. The inaccuracy of modern digital meters does not exceed 0.1%.
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Fig. 9.1. SimpliFed block diagram of the SR 830 lock-in ampliFer.
Selective ampli@ers serve to measure temperature oscillations or as null detectors in compensation circuits. Their main features are sensitivity, selectivity, stability, inherent noise, and convenience of tuning. For example, the ampliFer PAR 124A (Princeton Applied Research) operates as a selective nanovoltmeter in a frequency range 1 Hz–100 kHz. The operating frequency is set by a three-decade digital switch. The quality factor Q, adjustable up to 100, deFnes the selectivity. The input impedance of the ampliFer is 100 M\. A 1:100 input transformer allows a 100-fold enhancement of the sensitivity. However, in this case the internal resistance of the source of the signal should not exceed 10 \. The output voltages, proportional to the input voltage, are a DC voltage (up to 10 V at a 1-k\ load) and an ampliFed AC voltage (up to 0:1 V at a 600-\ load). The device is usable also as a wide-band ampliFer or a lock-in ampliFer. The gain stability is better than 1%, so that the ampliFer may serve for direct measurements. Frequency meters measure the modulation frequency. Usually, the calibration of oscillators is not suIciently accurate, and one has to measure the frequency by an additional digital frequency meter. At low frequencies, it is preferable to determine the period of the oscillations, because in this case the desired accuracy is obtainable in a shorter time. An external frequency meter is unnecessary when a digital setting of the frequency or a digital frequency meter is incorporated into the oscillator. Oscilloscopes are used to observe electric signals and as indicators in compensation circuits. Modulation frequencies are relatively low, whereas the signals after ampliFcation are suIciently large. The requirements for the oscilloscopes are therefore quite moderate. Lock-in ampli@ers measure periodic signals of chosen frequency even much weaker than signals of other frequencies or noise. Their main features are the frequency range, sensitivity, stability, and convenience of tuning. For example, the SR 830 (Stanford Research Systems) dual-channel lock-in ampliFer employs digital signal processing and provides full-scale sensitivity 2 nV to 1 V (Fig. 9.1). The frequency range is 1 mHz to 100 kHz. Input impedance of each channel is 10 M\, with 25 pF in parallel. Gain accuracy is 1%, and the absolute phase er◦ ◦ ror is less than 0:01 . Orthogonality of the channels is 90 ± 0:001 . The common-mode-rejection ratio at 100 Hz is 90 dB. The time constant is adjustable up to 30 s for frequencies above 200 Hz and up to 30 000 s for lower frequencies. An internal oscillator provides frequencies in the operating range. The inaccuracy in the frequency is ±25 ppm ± 3 × 10−5 Hz. The output impedance of this source is 50 \, the maximum voltage on a 50-\ load is 2:5 V (RMS). The
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Fig. 9.2. Block diagram showing electronic instrumentation of a calorimetric setup designed by Finotello et al. (1997). H—heater, T—thermometer.
results of the measurements are displayed, in digital form, as X and Y or R and . Here X is the component of the signal coinciding in phase with the reference, Y is the out-of-phase component, R is the modulus of the signal, and is the phase angle between the signal and the reference. The noise immunity of a lock-in ampliFer is limited by the so-called dynamic reserve of the ampliFer. This means the capability of measuring weak signals of a proper frequency in the presence of much stronger noise or periodic signals of other frequencies. The dynamic reserve of the ampliFer SR 830 is more than 100 dB. In some setups, calibrated variable resistors and capacitors are needed. Their inaccuracy usually does not exceed 0.1%. Photoelectric sensors detecting temperature oscillations include photomultipliers, photodiodes, and photoresistors. Their main features are the spectral response, stability, time constant, and inherent noise. Photomultipliers are very sensitive to visible and near-infrared radiation but are inconvenient because of a high operating voltage. Photodiodes and photoresistors provide a more favorable spectral response and are capable of detecting radiation from samples of lower temperatures. Data-acquisition systems serve to control the measurements and accumulate the experimental data. As an example of the electronic instrumentation for modulation measurements, one may consider a setup described by Finotello et al. (1997). The calorimetric cell is weakly anchored to a regulated bath and provided with a resistive heater (Fig. 9.2). A low-frequency oscillator feeds the heater. A resistance thermometer measures the temperature oscillations in the cell. Scanning the mean temperature of the cell by gradually changing the temperature of the bath, the heat capacity of the sample is obtainable versus temperature. The temperature stability of the bath is of the order of 0:1 mK at room temperature and a few microkelvins at liquid helium temperatures. A commercially available AC resistance bridge or a temperature controller may serve for this purpose, as well as a homemade AC bridge. The authors stressed that the success in the temperature control rests on using a high-resolution fast-responding thermometer. The power dissipated by the thermometer is 100 –1000 times smaller than that by the heater.
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The DC component of the voltage drop across the thermometer is directly measured and averaged by a digital meter, yielding the mean temperature of the cell. A preampliFer providing an adjustable frequency band ampliFes the AC component. Then the signal proceeds to a lock-in ampliFer operating at a long time constant for better signal averaging. Several ways exist to obtain the reference signal for the ampliFer. Lock-in ampliFers operate also in the so-called 2f mode, by creating a double-frequency reference signal. The authors used the 2f mode and the TTL output of the oscillator driving the heater. The dual-channel lock-in ampliFer provides both in-phase and quadrature outputs. The amplitude and the phase of the temperature oscillations are measured by a digital meter and stored in a computer. The computer stores also the mean temperature of the sample and controls the temperature of the bath. 10. Accuracy of modulation measurements As in other cases, errors of modulation measurements fall into two categories: errors arising from di0erences between the theoretical model and experimental conditions, and instrumental errors arising from the inaccuracy of measuring instruments. The theoretical model of the modulation calorimetry includes the following assumptions. (1) The mean temperature is the same over the calorimetric cell, as well as the amplitude and phase of the temperature oscillations. This means that when a separate heater and a thermometer are used, the time of equilibration between them and the sample is much shorter than the period of the modulation. (2) The modulation frequency is suIciently high to meet the criterion of adiabaticity: changes in the heat losses from the calorimetric cell due to the oscillations of its temperature are much smaller than the oscillations in the heating power. Assumptions (1) and (2) thus pose opposing requirements for the modulation frequency. (3) The heat capacity of the heater and thermometer is much smaller than that of the sample. This requirement is not so strict because of the high resolution of modulation measurements. In some cases, the heat capacity of the sample was much smaller than that of the addenda. The theoretical model is well satisFed when a conducting sample is heated by an electric current, while the temperature oscillations are detected through the resistance of the sample or by radiation from it. In this case, requirement (3) is excluded and assumption (2) is easy to meet by increasing the modulation frequency. To exclude the cold-end e0ects, the measurements are performed only on a central portion of the sample, where axial temperature gradients are small. Radial temperature di0erences are insigniFcant because of the high thermal conductivity and=or small thickness of the samples. Therefore, requirement (1) is also satisFed with direct electrical heating. With other methods of heating, assumption (1) is met by decreasing the sample’s thickness and the modulated power. When the modulated power is fed into the sample, its mean temperature increases. This increase is much larger than the temperature oscillations and may result in temperature gradients inside the sample. To reduce the gradients, only a small modulated power should be supplied to the sample, while a furnace controls the mean temperature. This method of heating is especially important in studies of phase transitions where good temperature resolution is one of the main requirements.
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Table 10.1 Estimates of errors in modulation measurements Source of error Mean temperature Mass of the sample Modulation frequency Oscillations of heating power Direct electrical heating Induction heating Modulated-light heating Electron bombardment Separate heaters Temperature oscillations Supplementary-current method Third-harmonic method Equivalent-impedance method Photoelectric detectors Thermocouples Resistance thermometers Oscillations of sample length, resistance, and thermal EMF
Imprecision (%)
Inaccuracy (%) 0.01–1 0.1–5 ¡ 0:01
0.1
0.2–1 4–6 2–5 1–2 0.1
1 0.1 0.1 0.5 0.01– 0.1 0.01– 0.1
1 1 1 1 0.1–1 0.1–1
0.01– 0.1
0.1–1
The thickness of the samples of low thermal di0usivity usually does not exceed 0:2 mm. A small thickness satisFes assumption (1) but contradicts requirement (3). To exclude this contradiction, the heater and thermometer may be prepared as deposited thin Flms, 10 –100 nm thick. To decrease the modulation frequency, one can employ a nonadiabatic regime. The second way is to reduce AC heat losses by means of an active thermal shield. This method is usable under the radiative heat exchange. With other mechanisms of heat exchange, the modulation frequency should be chosen carefully. The simplest way is to Fnd a frequency range where the quantity !0 remains constant. This means that both requirements (1) and (2) are satisFed. It is diIcult to meet them simultaneously only in one case, when samples of low thermal di0usivity are studied under high pressures provided by a dense medium. SigniFcant corrections are necessary to determine the speciFc heat in such measurements. Thus, in almost all cases the experimental conditions of modulation calorimetry are in good agreement with the theoretical model. In modulation measurements of thermal expansivity, electrical resistance, thermopower, and spectral absorptance, it is much easier to achieve such agreement because there is no need for the adiabatic regime of the measurements. In modulation measurements, the main sources of errors are the determinations of the oscillations in the heating power and in the sample’s temperature. Accurate determinations of the heating power are feasible with direct electric heating or separate resistive heaters. With other methods of heating, the situation is more complicated. Thermocouples and resistance thermometers are the best tools for accurate measurements of the temperature oscillations. When using other methods (temperature dependence of the sample’s resistance, photoelectric sensors), the errors depend on the reliability of the data used or on the accuracy of calibration. Table 10.1 shows estimates of inherent errors of modulation measurements. The term ‘imprecision’ refers to di0erences of single data points from smoothed values whereas ‘inaccuracy’
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refers to total errors including random and systematic errors. Usually, only rough estimates of the inaccuracy are available. 11. Applications of modulation techniques Studies employing modulation techniques are very numerous and cannot be presented in this review completely. Examples picked here illustrate the most important features of the modulation techniques: (i) the capability of studying metals at extremely high temperatures; (ii) the unique temperature resolution, from about 10−6 K at liquid helium temperatures to 1 K at high temperatures; (iii) the high precision, of the order of 10−4 ; (iv) the capability of measuring the speciFc heat as a function of an external parameter, such as pressure or magnetic Feld; (v) the small mass of the samples, down to 10−6 g; and (vi) the possibility to vary the modulation frequency in a wide range and to search for relaxation phenomena in the speciFc heat. Therefore, only several subjects are presented below. First, the modulation measurements provided many data on equilibrium point defects in metals. Second, they are widely used in studies of phase transitions in solids, including ferro- and antiferromagnets and superconductors. Third, the modulation techniques are the best tool to search for relaxation phenomena in the speciFc heat. At the same time, many important subjects have not been included (ferro- and antiferroelectrics, liquid crystals, thin Flms, conFned systems, biological materials). I would like to apologize to the authors of these studies, in particular, to K. Ema, D. Finotello, C.W. Garland, I. Hatta, C.C. Huang, G.S. Iannacchione, S. Imaizumi, H. Yao, and their collaborators. 11.1. Equilibrium point defects in metals Frenkel (1926) predicted the formation of point defects in solids many years ago as follows. Some atoms in a crystal lattice acquire energies much larger than the mean energy. At high temperatures, the energy of such atoms may become suIcient for them to leave their regular sites in the lattice and occupy interstitial positions. A vacancy and an interstitial thus appear simultaneously, the so-called Frenkel pair. Later, it was shown how vacancies may be created without formation of interstitials: atoms leaving their lattice sites occupy positions at the surface or at internal imperfections of the crystal (Wagner and Schottky, 1930). Such vacancies are often called Schottky defects. The equilibrium vacancy concentration obeys the relation cv = exp(−GF =kB T ) = exp(SF =kB ) exp(−HF =kB T ) = A exp(−HF =kB T ) ;
(11.1)
where GF is the Gibbs free energy of vacancy formation, and HF and SF are the corresponding enthalpy and entropy. The entropy SF does not include the conFgurational entropy. It relates to changes in vibration frequencies caused by softening of the atom binding near vacancies. Frenkel believed that the melting of solids is a result of softness of the lattice caused by point defects. He concluded that equilibrium concentrations of point defects may reach values of the order of
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1%. The Frst experimental observations of vacancies in metals were made in the 1950s when an extra resistivity of quenched samples was discovered. Annealing at appropriate temperatures recovers the resistivity. Defect-induced resistivity was also found in measurements at high temperatures. Point defects strongly inEuence many physical properties of metals at high temperatures: mechanical properties, enthalpy, speciFc heat, thermal expansion, electrical resistivity, and positron annihilation. Many data show that the predominant equilibrium point defects in metals are vacancies. The vacancy mechanism is the most probable one for the self-di0usion. Point defects may appear under nonequilibrium conditions, due to deformation or irradiation. Various theoretical calculations of the formation enthalpies of point defects are in reasonable agreement. However, equilibrium concentrations of the defects depend also on the formation entropies, theoretical results for which are ambiguous. Methods of studying point defects in metals may be divided into three groups as follows. (1) Studies of equilibrium defects through physical properties of metals at high temperatures. Results of such measurements do not depend on the history of the samples and are in satisfactory agreement. However, properties of a hypothetical defect-free crystal are unknown and cannot be calculated precisely. It is therefore impossible to reliably separate point-defect contributions. (2) Studies of samples in which extra concentrations of point defects were created by quenching, deformation, or irradiation. The main disadvantage of the equilibrium measurements is completely excluded here because the samples are compared with well-annealed samples, defect concentrations in which are negligible. Unfortunately, it is diIcult to reveal the equilibrium defect concentrations. During quenching, many vacancies have time to annihilate or form clusters. Therefore, vacancy concentrations after quenching may appear much smaller than equilibrium ones. This discrepancy grows on approaching melting points. (3) Observations of relaxation phenomena caused by the point-defect equilibration: properties of samples at high temperatures are studied under such rapid temperature changes that the defect concentration cannot follow them. In this case, one obtains values corresponding to a defect-free crystal. The relaxation technique is capable of an unambiguous separation of the defect contributions to physical properties. It is commonly accepted now that equilibrium point defects are to be studied under equilibrium conditions. Criteria for the choice of a suitable physical property for such experiments are quite clear: (i) the magnitude of the defect contribution and reliability of separating it; (ii) the accuracy of the measurements; and (iii) knowledge of parameters entering relations between the contributions and concentrations of the defects. The reasons that the most suitable property is the speciFc heat are as follows. (1) From theoretical calculations, the high-temperature speciFc heat of a defect-free crystal depends weakly on temperature. (2) As a rule, the defect contributions are much larger than the inaccuracy of the measurements. (3) The defect contribution directly relates to the equilibrium defect concentration. (4) Measurements of the speciFc heat under rapid temperature oscillations, necessary to separate the defect contributions, are simpler than corresponding measurements of other properties. Modulation techniques appear to be very useful for studying equilibrium point defects in metals (Kraftmakher and Strelkov, 1966b, 1970; Kraftmakher, 1971b, 1977, 1994a, b, 1996a, b, 1997, 1998a, b, 2000).
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Fig. 11.1. SpeciFc heat of metals. Pb, Al—adiabatic calorimetry (Kramer and N[olting, 1972); W, Pt—modulation calorimetry (Kraftmakher and Strelkov, 1962; Kraftmakher and Lanina, 1965); Cr—drop method (Kirillin et al., 1967); Mo, Nb—dynamic calorimetry (Cezairliyan et al., 1970; Righini et al., 1985). The di0erence between lowand high-melting-point metals is clearly seen.
11.1.1. Speci@c heat In the 1960s, the speciFc-heat data obtained by the equivalent-impedance technique were used to evaluate the enthalpies and entropies of vacancy formation in refractory metals. There exists a more or less extensive temperature range where the speciFc heat increases linearly with temperature. By extrapolating this dependence, the nonlinear contribution was separated and attributed to point-defect formation (Kraftmakher and Strelkov, 1962; Kraftmakher, 1963a, b, 1964, 1966c). These data gained no recognition at that time, as well as the results on tantalum and molybdenum by Rasor and McClelland (1960b). Due to the results of drop calorimetry, the opinion has been established that a linear temperature dependence of the speciFc heat continues up to the melting points of metals. Now the nonlinear increase in the high-temperature speciFc heat of metals is evident (Fig. 11.1). The excess molar enthalpy and speciFc heat caused by the vacancy formation are TH = NHF cv = NHF A exp(−HF =kB T ) ;
(11.2a)
TC = (NHF2 A=kB T 2 ) exp(−HF =kB T ) ;
(11.2b)
where N is the Avogadro number. A plot of ln T 2 TC versus 1=T should be a straight line, with a slope −HF =kB . After determination of the formation enthalpy, the coeIcient A and the equilibrium vacancy concentration become available. For tungsten, tantalum, molybdenum, and niobium, the parameters of equilibrium vacancies were deduced from the speciFc-heat data. A more rigorous determination of the vacancy-formation parameters consists in Ftting the experimental data by an equation taking into account vacancy formation: C = a + bT + cT −2 exp(−HF =kB T ) :
(11.3)
The coeIcients of this equation are obtainable using the least-squares method. All possible values of HF are tried, and for each value the standard deviation of experimental points is determined. A plot of the standard deviation versus the assumed value of HF shows the most probable formation enthalpy and its uncertainty.
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Table 11.1 High-temperature speciFc heat of metals from modulation measurements. The approximation takes into account the vacancy formation: C = a + bT + cT −2 exp(−d=T ) J mol−1 K −1 Metal
a
b × 105
c × 10−10
d × 10−3
Reference
W Ta Mo Nb Rh Zr Pt Ti Ni Cu Au La
20.5 24.35 22.05 20.7 22.5 20.5 24.5 24.05 27.2 23.65 23.85 24.7
545 285 650 525 91 420 545 440 530 500 525 650
740 220 170 30 80 35 25 46 50 5 2.5 37
36.5 33.6 26 23.7 22.05 20.3 18.6 18 16.2 12.2 11.6 11.6
Kraftmakher and Strelkov (1962) Kraftmakher (1963a) Kraftmakher (1964) Kraftmakher (1963b) Glazkov (1988) Kanel’ et al. (1966) Kraftmakher and Lanina (1965) Shestopal (1965) Glazkov (1987) Kraftmakher (1967c) Kraftmakher and Strelkov (1966a) Akimov and Kraftmakher (1970)
Table 11.1 presents approximation polynomials Ftting the high-temperature speciFc heats of metals obtained in modulation measurements. 11.1.2. Thermal expansion Vacancy formation causes an increase in the sample’s volume. After creation of a vacancy, a relaxation of the lattice occurs. The vacancy volume is thus smaller than the atomic volume 6 and equals 76 (7 ¡ 1). In an isotropic case, the changes in the volume, TV=V , and in the linear thermal expansivity of the sample, T#, are as follows: TV=V = 7cv = 7A exp(THF =kB T ) ;
(11.4a)
T# = (7HF A=3kB T 2 ) exp(−HF =kB T ) :
(11.5a)
The relative increase in the linear expansivity, T#=#, is much larger than that in the volume, TV=V . Hence, modulation measurements of the expansivity are preferable. The results of such measurements (Kraftmakher, 1967b, 1972) are in reasonable agreement with the relation (11:4b). However, the derived vacancy concentrations appeared somewhat smaller than those obtained from the speciFc-heat data. Besides an uncertainty in 7 values (they were assumed to be 0.5), there exists another reason for this disagreement. The main sources and sinks for vacancies are internal imperfections in the crystal lattice (voids, grain boundaries, dislocations and, probably, vacancy clusters). Therefore, vacancy formation may partly occur without an increase of the outer volume of the sample. The strong nonlinear increase of the expansivity of high-melting-point metals was observed by various techniques and now causes no doubts (Fig. 11.2). Di0erential dilatometry is commonly considered as being the most reliable method for determining equilibrium vacancy concentrations. This technique consists of simultaneously measuring the macroscopic dilatation, Tl=l; and the relative change in the lattice parameter, Ta=a, versus temperature. These quantities coincide at low temperatures. Due to vacancy formation,
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Fig. 11.2. Linear thermal expansivity of high-melting-point metals. Pt—modulation method (Kraftmakher, 1967b); Ir—traditional dilatometry (Halvorson and Wimber, 1972); Nb, Ta—dynamic techniques (Righini et al., 1986a; Miiller and Cezairliyan, 1982); W—recommended values (White and Minges, 1997).
a di0erence arises between the two quantities that grows rapidly with temperature. Under certain assumptions, the following relation is valid in an isotropic case: cv − ci = 3(Tl=l − Ta=a) ;
(11.5)
where ci is the equilibrium concentration of interstitials that is considered much smaller than the equilibrium vacancy concentration cv . This expression was used to determine vacancy concentrations in many metals. In all the cases, low concentrations were obtained, smaller than 0.1% at the melting points. However, such measurements have not been performed on high-melting-point metals. There are only a few data on the lattice parameter of high-melting-point metals at high temperatures, and further measurements are very desirable. 11.1.3. Electrical resistivity Point defects cause an additional scattering of conduction electrons. The extra resistivity due to vacancies, T*, is proportional to their concentration: T* = *v cv = *v A exp(−HF =kB T ) ;
(11.6)
where *v is a proportionality coeIcient. The inEuence of point defects on resistivity is similar to that of impurities. As a rule, the values of *v for various metals range from 1 to 10 \ cm per 1% vac. The inEuence of interstitials is several times larger. However, signiFcant deviations from Matthiessen’s rule were observed, i.e., the coeIcient *v depends on temperature. This must be taken into account when comparing extra resistivities from equilibrium and quenching experiments. The vacancy-induced extra resistivity is relatively small. Its separation therefore strongly depends on the extrapolation of the data from intermediate temperatures. The situation becomes more favorable when one directly measures the temperature derivative of the resistivity. For equilibrium vacancies, the increase of this derivative is T(d*=dT ) = (*v HF A=kB T 2 ) exp(−HF =kB T ) :
(11.7)
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Modulation measurements of the temperature derivative of resistance were carried out on aluminum and platinum (Kraftmakher and Sushakova, 1972, 1974). At intermediate temperatures, the derivative changes linearly with temperature, in accordance with the quadratic dependence of the resistance. A signiFcant nonlinear increase was seen at high temperatures, from which it is easy to deduce the vacancy contribution. For both metals, plausible formation enthalpies were obtained. The extra resistivities are close to the results of some quenching experiments. Thus, for the two metals there is no contradiction between equilibrium and quenching experiments, despite the diIculties mentioned above. 11.1.4. Equilibrium concentrations of point defects Surprisingly, until today we have no reliable knowledge of equilibrium point-defect concentrations in metals. The question is one of the most important, especially for high-temperature applications. Two opposing viewpoints on equilibrium point defects in metals are as follows. (1) Defect contributions to physical properties of metals at high temperatures are small and cannot be separated from the e0ects of anharmonicity. The only suitable methods to study equilibrium vacancies are positron-annihilation spectroscopy, which provides the enthalpies of vacancy formation, and di0erential dilatometry, which provides the equilibrium vacancy concentrations. The equilibrium vacancy concentrations at melting points range from 10−4 to 10−3 . Reasonable values of the formation enthalpies deduced from the nonlinear increase in the high-temperature speciFc heat of metals are accidental, and the derived defect concentrations are improbably large, so that this approach is generally erroneous. (2) In many cases, the defect contributions to the speciFc heat of metals are much larger than the nonlinear e0ects of anharmonicity and can be separated without crucial errors. This approach is quite adequate for determination of the defect parameters, especially, the equilibrium vacancy concentrations. These concentrations at melting points are of the order of 10−3 in low-melting-point metals and of 10−2 in high-melting-point metals. The strong nonlinear e0ects in the high-temperature speciFc heat and thermal expansivity of metals are caused by point-defect formation. Examination of these e0ects rules out anharmonicity as a possible origin of this phenomenon. Important arguments supporting this viewpoint have appeared in the last decade. It may turn out that calorimetric determinations provide the most reliable data on the equilibrium vacancy concentrations in metals. The equilibrium vacancy concentrations from di0erential dilatometry and from the calorimetric measurements are in strong contradiction (Fig. 11.3). The parameters of vacancy formation in metals derived from modulation measurements of speciFc heat are given in Table 11.2. Recently, the author reviewed the problem (Kraftmakher, 1998a, 2000). 11.2. Phase transitions in solids 11.2.1. First- and second-order phase transitions In studies of phase transitions in solids, speciFc-heat measurements answer two important questions, namely: (i) is the transition either of the Frst or the second order, and (ii) what is the temperature dependence of the speciFc heat near the transition point. Usually, modulation calorimetry does not allow measurements of the latent heat of a Frst-order phase transition. This feature may be considered as being an advantage because speciFc heats of both phases are not
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Fig. 11.3. Equilibrium vacancy concentrations in metals at melting points obtained from calorimetric measurements ( ) and from di0erential dilatometry ( ). The concentrations are of the order of 10−3 in low-melting-point metals and of 10−2 in high-melting-point metals. Note new di0erential-dilatometry data ( ) on copper and silver (Kluin and Hehenkamp, 1991; Mosig et al., 1992).
◦
•
Table 11.2 Parameters of equilibrium point defects in metals derived from modulation measurements of high-temperature speciFc heat Metal
HF (eV)
SF =kB
cmp (%)
Reference
W Ta Mo Nb Rh Zr Pt Ti Ni Cu Au La
3.15 2.9 2.24 2.04 1.9 1.75 1.6 1.55 1.4 1.05 1.0 1.0
6.5 5.45 5.7 4.15 5.25 4.6 4.5 5.15 5.4 3.7 3.15 5.8
3.4 0.8 4.3 1.2 1.0 0.7 1.0 1.7 1.9 0.5 0.4 1.2
Kraftmakher and Strelkov (1962) Kraftmakher (1963a) Kraftmakher (1964) Kraftmakher (1963b) Glazkov (1988) Kanel’ et al. (1966) Kraftmakher and Lanina (1965) Shestopal (1965) Glazkov (1987) Kraftmakher (1967c) Kraftmakher and Strelkov (1966a) Akimov and Kraftmakher (1970)
plagued by the latent heat, as for other calorimetric techniques. This was already seen in the Frst such studies (Holland, 1963; Kraftmakher and Romashina, 1965; Zaitseva and Kraftmakher, 1965). However, Garnier and Salamon (1971) have determined a small latent heat accompanying the phase transition in chromium. In these measurements, the temperature oscillations involved both phases, and a plateau was found due to the latent heat of the transition. The absorption and emission of the latent heat during a cycle of the temperature oscillations reduce the temperature changes (Fig. 11.4). Saruyama (1992) used modulation calorimetry to study the melting transition of an n-paraIn, C20 H42 . Mizuno et al. (1992) studied the ferroelectric phase transition in vinylideneEuoride-triEuoroethylene (VDF-TrFE) random copolymers. In the 52% VDF copolymer, they observed a frequency-dependent peak close to the transition point. The transition shows a heat-capacity peak even after excluding the Frst-order component. The authors explained this result as evidence that the transition has the nature of a high-order
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Fig. 11.4. Determination of a small latent heat of the phase transition in chromium, schematically (Garnier and Salamon, 1971).
transition, along with a latent heat. Ema et al. (1993) have pointed out that in such measurements the speed of the phase transition should be faster than the AC heating rate and superheating (or supercooling) should not be signiFcant. Such conditions are unlikely to be fulFlled in most Frst-order transitions in insulating materials. A more straightforward and adequate method is provided by the modulated di0erential scanning calorimetry (MDSC). This technique combines the advantages of modulation calorimetry and of di0erential scanning calorimetry. Hatta and Nakayama (1998) used modulation calorimetry and MDSC for studies of Frst-order phase transitions. They measured the speciFc heat of BaTiO3 and NaNO2 at their Frst-order ferroelectric-to-paraelectric phase transitions. In the MDSC measurements, the underlying heating rate was 0:1 K min−1 and the amplitude of the temperature oscillations was 0:1 K for periods of 100 and 60 s. In addition, the speciFc heat of octyloxycyanobiphenyl was determined by both techniques in a range including the Frst-order phase transition from the crystalline to the smectic-A phase. By comparison of the results obtained by MDSC with those by modulation calorimetry, the conclusions by the authors were as follows. (1) Heat capacity measurements at a Frst-order phase transition should be performed with temperature oscillations as small as possible. From such measurements, one can obtain the detailed behavior of the heat capacity except for the latent heat. (2) The latent heat can be determined, with high accuracy, from MDSC measurements. (3) When studying the kinetics of a phase boundary, the MDSC measurements should be carried out under conditions excluding a cooling process during the measurements. Fritsch et al. (1982, 1984) and Fritsch and L[uscher (1983) studied the speciFc heat of sodium and gallium very close to their melting points. Table 11.3 lists some studies of phase transitions in solids. 11.2.2. Second-order magnetic transitions In studies of second-order phase transitions, modulation calorimetry provides unique temperature resolution and sensitivity. Both features are crucial for precise determinations of the
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Table 11.3 Phase transitions in solids Item
Reference
Ti, #– transition CuZn SrTiO3 NH4 Cl RbAg4 I5 MAg4 I5 (M = Rb; K; NH4 ) 2H-TaSe2 ; 2H-TaS2 SnTe; Snx Ge1−x Te TiSe2 NH4 Br x Cl1−x C24 Rb intercalate Co, Zr CsBi(MoO4 )2 Pb1−x Gex Te (CH3 NH3 )2 FeCl4 Pt6 (NH3 )10 Cl10 (HSO4 )4 C6 Li intercalate CsPbCl3 NaN3 CuV2 S4 (Cn H2n+1 NH3 )2 FeCl4 (n = 1; 2) Quartz, #– transition Graphite-ICl intercalates AgCrS2 C60 R1:85 Ce0:15 CuO4−8 (R = rare earth) ZrTe3 K2 ZnCl4 ; K2 SeO4 ; Rb2 ZnCl4 CuGeO3 Cs2 ZnI4 C70 (NH4 )2 HPO4 ; (ND4 )2 DPO4 LiNH4 SO4
Holland (1963), Zaitseva and Kraftmakher (1965) Ashman and Handler (1969), Simons and Salamon (1971) Garnier (1971), Hatta et al. (1977) Schwartz (1971) Lederman et al. (1976), Jurado et al. (1997) Vargas et al. (1976, 1977) Craven and Meyer (1977) Hatta and Kobayashi (1977), Hatta and Rehwald (1977) Craven et al. (1978a) Lushington and Garland (1980), Yoshizawa et al. (1983) Suematsu et al. (1980) Boyarskii and Novikov (1981) Stokka and Samulionis (1981) Sugimoto et al. (1981) Goto et al. (1982) Inoue et al. (1982) Robinson and Salamon (1982) Stokka et al. (1982) Hirotsu et al. (1983) Sekine et al. (1984) Yoshizawa et al. (1984) Matsuura et al. (1985), Yao and Hatta (1995) Tashiro et al. (1985, 1990) Kawaji et al. (1989) Chung et al. (1992) Hwang et al. (1992) Chung et al. (1993b) Haga et al. (1995a, b) Kuo et al. (1995, 1996) Melero et al. (1995) Sekine et al. (1995) Vargas et al. (1995), Vargas and Diosa (1997) Diosa et al. (2000)
temperature dependence of speciFc heat. In this case, absolute values of the speciFc heat are not so important, and all variants of modulation calorimetry are usable. The theory of the second-order phase transitions predicts a power-law temperature dependence of the speciFc heat near transition points: −
C − (t) = (A− =#− )(|t |−# − 1) + B− +
C + (t) = (A+ =#+ )(t −# − 1) + B+
for T ¡ TC ; for T ¿ TC ;
(11.8a) (11.8b)
where t = T=TC − 1; TC is the transition temperature, and #− and #+ are the so-called critical indices of the speciFc heat for T ¡ TC and T ¿ TC , respectively.
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Fig. 11.5. Determination of the transition temperature and of the critical indices of the speciFc heat, schematically (Connelly et al., 1971).
Generally, A+ and A− do not coincide, as well as B+ and B− . Experimental data below and above the transition point are treated separately, and di0erent values of TC Ft the two sets of data. Usually, at the Fnal stage one accepts Fts with the same TC for both branches of the speciFc heat. An additional requirement often posed is the same critical index below and above the transition point, i.e., #+ = #− . Connelly et al. (1971) used this procedure to derive the transition temperature and the critical indices for nickel. The dependence of #+ and #− on TC allowed the authors to Fnd the point where #+ =#− . This temperature, accepted as the transition temperature, was about 0:13 K above the point of the maximum in the speciFc heat. Taking into account the uncertainties in the parameters Ftting the experimental data, a region of probable values of TC and # appears (Fig. 11.5). The authors concluded that #+ = #− = −0:10 ± 0:03. The Frst modulation measurements of the speciFc heat of ferromagnets near the Curie points were made in the 1960s on iron (Kraftmakher and Romashina, 1965), nickel (Kraftmakher, 1966a), and cobalt (Kraftmakher and Romashina, 1966). The singularity in the Curie point was found to be of logarithmic type, which means that #+ = #− = 0. The coeIcients of the polynomials Ftting the data for the three ferromagnets are given in Table 11.4. This approximation means that the speciFc heat behaves symmetrically below and above the transition point, but the high-temperature branch is shifted down. However, only upper and lower limits for the critical indices are obtainable from experimental data. In the Frst measurements, it was concluded that |#| ¡ 0:2. Ikeda et al. (1976) observed a symmetric logarithmic singularity in the speciFc heat of the two-dimensional Ising-like antiferromagnets K2 CoF4 and Rb2 CoF4 near their NReel points. Ikeda et al. (1981) reported a similar singularity for the two-dimensional random antiferromagnets
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Table 11.4 SpeciFc heat of iron, cobalt, and nickel near their Curie points. The experimental data are Ftted by polynomials C ± =3NkB = −A log |T − TC | + B± , where A = A+ = A− Metal
A
B−
B+
TB
Iron Cobalt Nickel
1.65 0.7 0.46
9.15 6.95 4.72
8.15 6.3 4.4
1 0.65 0.32
Rb2 Cox Ni1−x F4 (x = 0; 0:5; 0:65, and 0:8). Hatta and Ikeda (1980) have found a symmetric logarithmic divergence of the speciFc heat of several antiferromagnets of the K2 NiF4 family. However, in many other cases the speciFc heat of ferro- and antiferromagnets manifests a more complex behavior. The speciFc-heat anomaly is not symmetric, and the critical indices depend on temperature. For instance, Lederman and Salamon (1974) observed a change of the critical indices of the speciFc heat of dysprosium near the NReel point. At |t | = 5 × 10−3 ; #+ = #− = −0:02 ± 0:01 for the outer region, while for the inner region #+ = #− = 0:18 ± 0:08. A precise determination of the critical indices is diIcult because of imperfection of the samples (impurities, grain boundaries, dislocations). This causes rounding of experimental points in the close vicinity of the transition temperature. The modulation microcalorimetry capable of measurements on very small samples probably would clarify the situation. In 1960s, a question arose about the temperature dependence of the electrical resistivity of a ferromagnet near its Curie point. The modulation technique immediately provides data on the temperature derivative of the sample’s resistance and thus is more suitable to answer this question (Kraftmakher, 1967a). It turned out that this derivative behaves like the speciFc heat, consisting of a jump and a power-law temperature dependence (Fig. 11.6). Fisher and Langer (1968) developed a theory that predicts such a behavior above the Curie point. Shacklette (1974) has examined this prediction more accurately. Terki et al. (1992) studied the resistivity of erbium near its NReel point. Park et al. (1997) determined the speciFc heat and electrical resistivity of the ferromagnet La0:7 Ca0:3 MnO3 . It was shown that below the Curie point the temperature coeIcient of the resistivity is proportional to the speciFc heat. This proportionality is valid also in magnetic Felds, up to 1 T. Modulation studies of phase transitions in ferro- and antiferromagnets are listed in Tables 11.5 and 11.6. 11.2.3. Superconductors Sullivan and Seidel (1968) carried out the Frst modulation studies on a superconductor. They measured the speciFc heat of indium versus an external magnetic Feld. The aim of these measurements was to demonstrate the important advantages of the new calorimetric technique. Since then, many studies of superconductors were performed, including bulk samples, thin Flms, Fne particles, organic superconductors, pure metals and alloys, crystalline and amorphous. Greene et al. (1972) measured the speciFc heat of granular Al–Al2 O3 Flms with millikelvin resolution using modulated-light heating. Farrant and Gough (1975) studied niobium in a vicinity of the transition point at constant magnetic induction. The magnetic Feld was trapped inside the
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Fig. 11.6. Temperature derivative of resistance of nickel near the Curie point (Kraftmakher, 1967a).
sample by allowing it to cool from a temperature well above TC in a homogeneous magnetic Feld. Tsuboi and Suzuki (1977) measured the speciFc heat of tin particles whose average radius ranged from 13 to 110 nm. Thin oxide layers mutually insulated the particles. A substrate for the particles made of thin glass (5 –10 m) was mounted on a copper frame. A 20-nm manganin heater was deposited on the glass. A SiO Flm, 300 nm thick, was deposited on the heater, and then a 300-nm gold Flm to improve thermal di0usion was deposited over the region of the sample. A second SiO Flm served to insulate the gold Flm from a germanium thermometer. On the opposite side of the substrate, tin particles were deposited in vacuum on an area of 1 × 1 cm2 . After an amount of the particles of 0:5 mg or more was obtained, a 300-nm SiO Flm was deposited on the particles to protect them from oxidation in air. At 4:2 K, the contribution of the particles was at most 10% of the total heat capacity. The authors summarized their main results as follows. (1) The temperature of the maximum in the speciFc heat is lower than the transition temperature for bulk samples. (2) As the particle size decreases, the peak of the speciFc heat shifts towards lower temperatures and becomes broader. Similar measurements were performed under magnetic Felds up to 3 T (Suzuki and Tsuboi, 1977). Along with usual
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Table 11.5 Phase transitions in ferromagnets Item
Reference
Fe, speci@c heat
Kraftmakher and Romashina (1965), Lederman et al. (1974), Varchenko et al. (1978) Kraftmakher and Romashina (1967), Kraftmakher and Pinegina (1974), Shacklette (1974) Kraftmakher and Pinegina (1970) Kraftmakher (1966a), Handler et al. (1967), Connelly et al. (1971), Maszkiewicz (1978), Maszkiewicz et al. (1979), Ohsawa et al. (1978) Kraftmakher (1967a) Papp (1984) Kraftmakher and Romashina (1966) Kraftmakher and Pinegina (1971) Lewis (1970), Wantenaar et al. (1977), Jeong et al. (1991), Jung et al. (1992), Bednarz et al. (1992, 1993); Glorieux and Thoen (1994), Glorieux et al. (1995) Salamon and Simons (1973) Simons and Salamon (1974)
dR=dT Thermopower Ni, speci@c heat dR=dT SpeciFc heat and thermopower Co, speci@c heat Thermopower Gd Transition at 226 K SpeciFc heat and dR=dT Other ferromagnets, speci@c heat EuO CoS2 Fe75 P15 C10 , amorphous Fe34 Pd 46 P20 , amorphous Fe80−x Mx P13 C7 (M = Ni; Mn; Cr) Eux Sr 1−x S Hf 1−x Tax Fe2 , amorphous YCo12 B6 ; GdCo12 B6 (ferrimagnet) NdNi2 ; TbNi2 ; DyNi2 Ni90 Cr 10 La0:7 Ca0:3 MnO3 Thermopower Fex Co80−x B20 ; Fex Ni80−x B19 Si ErCo2
Salamon (1973) Ogawa and Yamadaya (1974), Hiraka and Endoh (1999) Schowalter et al. (1977) Craven et al. (1978b) Ikeda and Ishikawa (1980) Haeiwa et al. (1988) Murayama et al. (1995) Nahm et al. (1995) Melero and Burriel (1996) Jurado et al. (1997) Park et al. (1997) Kettler et al. (1982, 1984) Resel et al. (1996)
measurements of the speciFc heat versus temperature for several magnetic Felds, the authors carried out Feld-sweep measurements. Studies of low-temperature superconductors are listed in Table 11.7. SpeciFc-heat anomalies in high-temperature superconductors pose serious problems. First, the phonon speciFc heat near the transition points is large, and the contribution to be studied is of the order of 1%. Second, measurements on single crystals are desirable, and a small volume of the samples, smaller than 0:1 mm3 , causes additional diIculties. Modulation calorimetry is therefore very attractive for such studies. Inderhees et al. (1987) were the Frst to apply modulation calorimetry in a study of YBa2 Cu3 O7−8 (YBCO). The ceramic sample was a disc, 3 mm in diameter and 0:5 mm thick,
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Table 11.6 Phase transitions in antiferromagnets Item
Reference
Cr, speciFc heat and dR=dT CoO Cr K2 NiF4 MnBr 2 :4H2 O K2 MnF4 ; K2 NiF4 Dy K2 CoF4 ; Rb2 CoF4 CsMnF3 MnF2 Rb2 Cox Mg1−x F4 K2 NiF4 ; K2 CoF4 ; Ba2 CoF6 ; Rb2 NiF4 ; K2 MnF4 CoF2 ; MnF2 ; KMnF3 FeF2 Rb2 Cox Ni1−x F YFe2 O4 C6 Eu intercalate ZnCr 2 Se4 Mn0:5 Zn0:5 F2 GdCu2 Si2 ; GdNi2 Si2 ; GdGa2 ; GdCu5 Er; dR=dT CoO; Cr 2 O3 Cr 2 O3 ; FeF2 ; Cr Tb2 O2 SO4 , magnetic phase diagram Dy, magnetic phase diagram Gd 2 O2 SO4 , magnetic phase diagram R1:5 Ce0:5 Sr 2 FeCu2 O9 (R = Y; Eu)
Salamon et al. (1969) Salamon (1970) Garnier and Salamon (1971) Salamon and Hatta (1971) Hempstead and Mochel (1973) Salamon and Ikeda (1973) Lederman and Salamon (1974) Ikeda et al. (1976) Ikeda (1977) Ikeda et al. (1978) Suzuki and Ikeda (1978) Hatta and Ikeda (1980) Akutsu and Ikeda (1981) Hatta and Ikushima (1981) Ikeda et al. (1981) Tanaka et al. (1982) Ohmatsu et al. (1983) Ershov et al. (1984) Ikeda (1986) Bouvier et al. (1991) Chaussy et al. (1992), Terki et al. (1992) Glorieux et al. (1994) Marinelli et al. (1994) Thurner et al. (1995) Izawa et al. (1996) Kratz et al. (1996) Felner et al. (1997)
with a mass of 18 mg. Modulated light provided the AC heating. A Eattened thermocouple, made of 25-m chromel and constantan wires, was attached to the backside of the sample with a small amount of varnish. The temperature oscillations, of about 5 mK, were measured and recorded by means of a lock-in ampliFer and a computer. The same thermocouple recorded the enhancement of the mean temperature. A calibrated carbon-glass thermometer measured the temperature of the bath. Adiabatic heat-pulse measurements at 77 K were carried out to normalize the data. The electronic speciFc heat of YBCO was calculated from the Pauli paramagnetic susceptibility. The lattice contribution was Ftted according to the Debye formula. The ratio TC=7TC was found to be 1:23 ± 0:08, which is close to the value of 1.43 predicted by the Bardeen–Cooper–Schrie0er theory. Later, this group has performed calorimetric measurements on small single crystals, including measurements in magnetic Felds (Salamon et al., 1988, 1990; Inderhees et al., 1988, 1991; Ghiron et al., 1992). Chung et al. (1996) measured the speciFc heat of single whiskers of BiSCCO in the range 25 –280 K. The sample had a typical size of 2 × (0:2–0:3) × (0:01–0:03) mm3 with a calculated mass from 40 to 100 g. One of the large surfaces of the sample was exposed to a light
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Table 11.7 SpeciFc heat and thermopower of low-temperature superconductorsa Item Speci@c heat In, in magnetic Feld InPb alloys BiSb, amorphous Flms
Reference
Al, granular Flms Ag–Pb–Ag sandwiches V3 Ga; Nb3 Al Nb, in magnetic Feld Al foiles, Sn Flms LaRu2 Sn, Fne particles In Flms V3 Si, in magnetic Feld Zr 3 Rh, amorphous Bi84 Sb16 and Ga95 Ag5 Flms Nb3 Ge Flms Al, thin Flms (TMTSF)2 ClO4 , in magnetic Feld CuZr, amorphous BaPb1−x Bix O3 Sn1−x Cux , amorphous (-(BEDT-TTF)2 Cu(NCS)2 Al Ba0:6 K0:4 BiO3 Aux Sn1−x -(BEDT-TTF)2 I3 PtGa2 Cux Sn1−x
Sullivan and Seidel (1968) Zoller and Dillinger (1969) Zally and Mochel (1971, 1972), Krauss and Buckel (1975) Greene et al. (1972) Manuel and VeyssiRe (1973) Viswanathan et al. (1974) Farrant and Gough (1975) Manuel and VeyssiRe (1976) Viswanathan et al. (1976) Suzuki and Tsuboi (1977), Tsuboi and Suzuki (1977) Gibson et al. (1979) Huang et al. (1980) Garoche and Johnson (1981) K[ampf et al. (1981) Rao and Goldman (1981) Suzuki et al. (1982) Brusetti et al. (1983), Fortune et al. (1990) Garoche and Bigot (1983) Sato et al. (1983) Dutzi and Buckel (1984) Katsumoto et al. (1988), Graebner et al. (1990) Machado and Clark (1988) Graebner et al. (1989) Rieger and Baumann (1991) Fortune et al. (1992) Hsu (1994) Sohn and Baumann (1996)
Thermopower LaRu2
Resel et al. (1996)
a
TMTSF = tetramethyltetraselenafulvalene; BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene.
beam chopped at a frequency typically below 10 Hz. This surface was coated with an evaporated 0:1-m PbS Flm to eliminate e0ects of a temperature-dependent absorptivity. A 13-m thermocouple attached to the opposite large surface of the sample with diluted GE 7031 varnish measured the temperature oscillations. A broad maximum in the C=T curve was observed because the whiskers contained a mixture of two superconducting phases. Carrington et al. (1996) employed a di0erential modulation calorimeter in studies of a 30-g ∼ 15 K). The sample under study and a reference one single crystal of Tl2 Ba2 CuO6+8 (TC = were attached to chromel-constantan thermocouples and placed on opposite sides of a copper plate. They were heated independently, via optical Fbers, by two light-emitting diodes. It was possible to monitor the temperature oscillations of each sample or to measure the di0erence
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between them. At a temperature near the transition, this di0erence was adjusted to be close to zero. Measurements in magnetic Felds up to 7 T served to determine the normal background. ∼ 111 K), the sample was a small platelet crystal In measurements on HgBa2 Ca2 Cu3 O8+8 (TC = 3 with dimensions 0:2 × 0:1 × 0:03 mm and a mass of about 5 g (Carrington et al., 1997). The di0erential method greatly improved the sensitivity and reproducibility of the measurements and e0ectively eliminated the e0ect of the magnetic Feld on the thermocouple calibration. To obtain a reference sample with a phonon contribution well matched to the sample under study, ◦ one of the samples was annealed at 450 C. After this treatment, the sample had no traces of superconductivity above 60 K. Charalambous et al. (1999) studied the asymmetry of the critical indices in YBCO. A high-resolution microcalorimeter (Riou et al., 1997) was employed. The high precision of the calorimeter, of the order of 10−4 , allowed the authors to carry out the measurements on small detwinned single crystals. The asymmetry of the two branches of the speciFc heat, below and above the transition point, was clearly seen from the dC=dT data. The authors have concluded that the two critical indices are deFnitely di0erent: #− = −0:2 ± 0:3; #+ = 0:5 ± 0:2. Such a behavior cannot be explained within the framework of a regular second-order transition. Howson et al. (1989, 1990) measured the thermopower of YBCO crystals in the range from the transition point TC to 250 K and in magnetic Felds up to 2 T. The thermopower was measured using modulated-light heating relative to Pb reference leads. A sharp peak close to TC is very sensitive to the oxygen content of the samples. The authors argued that the peak may be due to Euctuation e0ects. Table 11.8 lists studies of high-temperature superconductors. 11.2.4. An unexpected premelting anomaly An unexpected anomaly was observed in the speciFc heat of platinum near its melting point (Kraftmakher, 1978b). Wire samples, 0.02–0:1 mm in diameter, were heated in vacuum by passing through them a DC current with a small AC component. The light from a small central portion of the sample was projected onto a photosensor, a photomultiplier or a photodiode. The modulation frequency was in the range 20–2000 Hz, while the temperature oscillations ranged from 0.1 to 1 K. Just before destruction of the samples, the oscillations in the sample’s radiation rapidly increased. Usually, the increase was 20 –30 times but in several samples it reached 50 times. To reduce the amplitude of the oscillations, it was necessary to decrease the AC component of the heating current. Under a gradual increase of the heating current, after reaching the point C (Fig. 11.7) and even at somewhat higher temperatures, it was possible to return to the point A and repeat such cycles several times. The destruction of the samples occurred in the region CD. The phenomenon was observable only in a central portion of the samples, shorter than 1 mm. It was impossible to directly determine the temperature region of the anomaly, and only rough estimates could be made. The estimates correspond to approximately 10 K. After the Frst observations, the phenomenon was conFrmed when platinum samples were heated by electron bombardment (Fridman, 1983). Further details were reported later (Kraftmakher, 1991). Unsuccessful attempts were undertaken to Fnd an anomaly in the electrical resistivity of the samples. The resistance of the entire sample was measured, so that a contribution from a small portion of the sample could not be
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Table 11.8 SpeciFc heat and thermopower of high-temperature superconductors Item
Reference
YBCO, speciFc heat
Inderhees et al. (1987, 1988), Ishikawa et al. (1988), Kishi et al. (1988), Anshukova et al. (1989), L]greid et al. (1989), Marinelli et al. (1990), Regan et al. (1991), Zhou et al. (1991), Kamilov et al. (1995), Marone and Payne (1997), Riou et al. (1997), GarFeld et al. (1998), Maesono and Tye (1998) Ginsberg et al. (1988), Calzona et al. (1990), Salamon et al. (1988, 1990, 1993), Ghiron et al. (1992), Overend et al. (1994), Overend et al. (1996) Inderhees et al. (1991) L]greid et al. (1988), Vargas et al. (1989) Lowe et al. (1991) Yu et al. (1988), Howson et al. (1989, 1990), Oussena et al. (1992)
In magnetic Feld
Along Hc2 line Transitions at 90 and 220 K YBCO, thermopower In magnetic Feld Other HTSCs, speci@c heat La1:85 Sr 0:15 CuO4 Bi0:7 Pb0:3 SrCaCu1:8 Ox (Bi,Pb) –Sr–Ca–Cu–O Bi2 Sr 2 CaCu2 Oy Bi1:7 Pb0:3 Sr 2 Ca2 Cu3 O10−8 Bi2 Sr 2 Can−1 Cun Ox whiskers (n = 1; 2) Tl2 Ba2 CuO6+8 , in magnetic Feld HgBa2 Ca2 Cu3 O8+8 , in magnetic Feld La2 CuO4:093 Thermopower Sr 0:15 La0:85 CuO4−8 , in magnetic Feld Nd 1:85 Ce0:15 CuO4+y Nd 2−x Cex CuO4 Tl2 Ba2 CuO6
Feng et al. (1988) Slaski et al. (1989) Okazaki et al. (1990) Nes et al. (1991) Ausloos et al. (1994) Chung et al. (1996) Carrington et al. (1996) Carrington et al. (1997) Hirayama et al. (2000) Yu et al. (1988) Mangelschots et al. (1992) Xu et al. (1992) Lin et al. (1993)
detected. However, by measuring the oscillations of the radiation from the entire sample the anomaly was still observable: the oscillations exhibited an increase by 1.5 –2 times. The increase of the oscillations thus cannot be a simple result of a local increase of the resistivity. A possible origin of the phenomenon might be an anomaly in the emittance of the samples. To check this conjecture, temperature oscillations of two frequencies, low and infralow, were created in the samples simultaneously. It turned out that in the region where the low-frequency oscillations of the radiation rapidly increase, the infra-low oscillations (0.1 and 0:2 Hz) exhibit no anomaly. In addition, a change in the phase of the temperature oscillations was observed at 20 Hz. This conFrms that the phenomenon is caused by an anomaly in the speciFc heat rather than in the emittance of the sample’s surface.
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Fig. 11.7. Unexpected premelting anomaly in speciFc heat of platinum and PtRh alloy, schematically (Kraftmakher, 1991).
The results described above were obtained on samples 20 mm long. To lengthen the region of the anomaly, longer samples were also investigated. However, in this case the destruction of the samples occurred earlier, in the region BC. Probably, the equilibrium melting corresponds to the point B, while the anomaly relates to a superheated (metastable) state. Such a conclusion was made because the speciFc heat in the anomalous region is too small to be attributed to any equilibrium solid or liquid state. The strong temperature dependence of the speciFc heat in the anomalous region causes signiFcant distortions of the temperature oscillations. The distortions were observable directly by an oscilloscope and by detecting the second-harmonic signal. Probably, Bezemer and Jongerius (1976) observed just this phenomenon when determining the melting point of platinum. The authors attributed the distortions to changes in the emittance of the sample during melting and did not consider the speciFc heat. The anomaly was also detected by the equivalent-impedance technique, which provides the ratio of the heat capacity of the sample to the temperature derivative of its resistance. An anomalous behavior of this quantity was observed just before the destruction of the samples. However, such measurements relate to the entire sample, while the anomaly relates to a small portion of it. In PtRh alloys (10% and 30% Rh), the anomaly is more moderate and its temperature region seems to be wider. The phenomenon was observed also in nickel and palladium, whereas no anomaly was seen in tungsten, tantalum, and niobium. 11.3. Temperature Ductuations and isochoric speci@c heat of solids Observations of temperature Euctuations under equilibrium conditions o0er a unique opportunity to determine the isochoric speciFc heat of solids (Kraftmakher and Krylov, 1980a, b). The mean square of the temperature Euctuations in a sample, TT 2 , and their spectral density, TTf2 , are related as follows: TT 2 = kB T 2 =mCv ;
(11.9a)
TTf2 = 4kB T 2 =P (1 + x2 ) ;
(11.9b)
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Fig. 11.8. Measurement of small temperature oscillations and of temperature Euctuations at high temperatures (Kraftmakher and Krylov, 1980a).
where kB is Boltzmann’s constant, P is the heat transfer coeIcient, x = mCv !=P , and m and Cv are the mass and isochoric speciFc heat of the sample, respectively (Milatz and Van der Velden, 1943; Landau and Lifshitz, 1980). It is very important that the isochoric speciFc heat Cv , rather than the isobaric speciFc heat Cp , enters the above expressions. The reason is that the temperature Euctuations and Euctuations of the sample’s volume are uncorrelated. In contrast to liquids, it is impossible to directly measure the isochoric speciFc heat of solids. Measurements of the temperature Euctuations provide such an opportunity. The main diIculty is the smallness of the temperature Euctuations, even in very small samples. To observe markedly di0erent values of the two speciFc heats, Cp and Cv , the measurements are to be performed at high temperatures. Photoelectric sensors are the best tools to detect extremely small temperature oscillations and temperature Euctuations at high temperatures. A simple setup was built for the measurements. The signal was expected to be comparable with the inherent noise of a photosensor. To suppress this noise, a correlation method was used (Fig. 11.8). The setup includes two identical channels, each consisting of a photodiode, a preampliFer, and a power ampliFer. An electrodynamometer serves to accurately multiply the signals fed into its movable coil and one of the Fxed coils. The displacement of the movable coil is registered by means of an additional low-frequency current passed through it. A voltage induced in the second Fxed coil depends on the orientation of the movable coil. This voltage is measured by a lock-in ampliFer with a long time constant and then recorded. An averaging of the signal over several hours thus is possible. When detecting small temperature oscillations in wire samples, a low-frequency oscillator performs the modulation of the heating power. Only one ampliFcation channel is employed, while the oscillator supplies the second signal for the multiplier. In this case, the electrodynamometer operates as a lock-in detector. Using the setup, temperature oscillations in the range 10−6 –10−5 K were measurable. Probably, there is no chance to accurately measure all the quantities entering the above formulas and to deduce the isochoric speciFc heat Cv . However, modulation calorimetry solves the problem. When the same sample is subjected to a modulated heating power, the temperature oscillations in it are governed by the isobaric speciFc heat Cp . This allows one to exclude all the quantities that cannot be accurately measured and to determine the speciFc-heat ratio Cp =Cv .
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•
Fig. 11.9. Spectra of temperature Euctuations ( ) and of temperature oscillations caused by noise modulation of heating power (· · · · ·). The spectra Ftted at low frequencies show the Cp =Cv ratio (Kraftmakher and Krylov, 1980b). Fig. 11.10. Determination of Cp =Cv ratio for tungsten using the least-squares method.
To deal with spectral densities in both cases, a noise generator performs the modulation, and the temperature oscillations in the sample are of the same nature as the temperature Euctuations. The only di0erence is that they are governed by the isobaric speciFc heat. The mean temperature of the sample is the same, and the measurements are carried out using the same measuring system. To determine the speciFc-heat ratio, it is suIcient to compare the frequency dependence of the temperature Euctuations and that of the oscillations caused by the noise modulation. When the modulation is performed by means of a noise generator, the spectral density of the temperature oscillations in the sample is given by 2f = pf2 =P 2 (1 + y2 ) ;
where pf2 is mCp !=P . The
(11.10)
the spectral density of the square of the heating-power oscillations, and y = ratio of the two spectral densities can be written in the form
TTf2 = 2f = A(1 + z 2 )=(1 + z 2 =72 ) :
A = 4kB P = pf2
(11.11)
Here is constant at a given temperature, z=f=f0 ; f0 is the modulation frequency √ at which the temperature oscillations become 2 times smaller than at zero frequency, and 7 = Cp =Cv . Eq. (11.11) makes it possible to determine the Cp =Cv ratio by the least-squares method. Such a determination is feasible even without an evaluation of the temperature oscillations, the mass, and the heat transfer coeIcient of the sample. The only requirement is that the spectral density of the noise used for the modulation be independent of the frequency. Temperature Euctuations were observed in thin tungsten wires, 3:5 m thick and 1 mm long. At 2200 and 2400 K, the spectra appeared in good agreement with the theoretical prediction (Fig. 11.9). Below 20 Hz, the spectral density of the temperature Euctuations is constant and equals approximately 3 × 10−11 K 2 Hz−1 , whereas at high frequencies it is inversely proportional to the frequency squared. The Cp =Cv ratio was found to be 1:4 ± 0:1 (Fig. 11.10). The isochoric speciFc heat at 2200 K is 24 ± 2:5 J mol−1 K −1 . The electronic contribution was taken from
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low-temperature measurements (Waite et al., 1956) and believed to be linear with temperature. At 2200 K, it amounts to about 2:5 J mol−1 K −1 . The isochoric speciFc heat of the crystal lattice is thus somewhat smaller than the classical limit 3NkB = 24:9 J mol−1 K −1 . This result does not contradict the theory because the lattice speciFc heat may decrease with increasing temperature because of a negative anharmonic contribution. Further determinations of the Cp =Cv ratio unquestionably deserve e0orts. Nowadays, a data-acquisition system and a computer could process the data. Chui et al. (1992) observed temperature Euctuations at low temperatures. The authors pointed out that the central point of the problem is the applicability of the relation between Euctuations of two fundamental quantities in thermodynamics, the energy U and the temperature T . One point of view consists in accepting the relation TU = CTT , where C is the heat capacity of a subsystem thermally connected to a reservoir. Energy exchanges between them lead to the above formulas for the temperature Euctuations. Another approach rests on the statement that the temperature of a canonical ensemble is constant and does not Euctuate. Temperature Euctuations in a microcanonical ensemble are possible but in this case TT 2 = kB T 2 =MC, where MC is the total heat capacity of the system and the reservoir. The experiment was aimed at a choice between the two theories of temperature Euctuations. The authors measured the temperature-dependent magnetization of a paramagnetic salt, copper ammonium bromide, in a Fxed magnetic Feld. A SQUID coupled to a superconducting coil wound around the salt pill detected the changes in the magnetization. The temperature Euctuations were measured at temperatures near 2 K, above the Curie point of the salt. To exclude another origin of the phenomenon, Euctuations in two samples were measured simultaneously, and no correlation between them was found. Due to the high sensitivity of the SQUID measurements of the spectral densities of the order of 10−20 K 2 Hz−1 were possible. The spectral density of the temperature Euctuations appeared in agreement with expectations based on the known heat capacity and the determined relaxation time of the samples. Below 0:1 Hz, the spectral density was about 10−19 K 2 Hz−1 . Estimates were made of the Euctuations in the magnetization of the samples, which could be erroneously considered as being the temperature Euctuations. This e0ect causes an e0ective noise much smaller than the Euctuations observed. The authors have concluded that the relation TU = CTT is applicable to the Euctuations in U and T to an accuracy of 20%. Since the heat capacity of the reservoir was 1700 times larger, the authors claimed that the results conFrm the Euctuation theory leading to the above formulas. Later, Day et al. (1997) discussed the Euctuation-imposed limit for temperature measurements. 11.4. Relaxation phenomena in the speci@c heat 11.4.1. Formulas for the relaxation Modulation calorimetry provides an opportunity to vary the frequency of the temperature oscillations in a wide range. This makes possible a search for relaxation phenomena, which appear when the modulation period becomes comparable to the relaxation time of a process contributing to the speciFc heat. Three such phenomena have been found by modulation calorimetry. First, supercooled organic liquids were studied near glass transitions. In this region, the speciFc heat becomes complex and frequency dependent (Birge and Nagel, 1985, 1987; Birge, 1986). Second, a relaxation phenomenon in the speciFc heat of tungsten and platinum was found at
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Fig. 11.11. Expected relaxation in speciFc heat versus X = !, for TC=C = 0:1. 1—change in speciFc heat, 2—shift in phase of temperature oscillations.
modulation frequencies of the order of 105 Hz (Kraftmakher, 1985, 1990). The phenomenon was attributed to equilibration of point defects in the crystal lattice. Finally, relaxation was observed in the low-temperature speciFc heat of Mn12 acetate crystal (Fominaya et al., 1997b, 1999b). To calculate the relaxation, Van den Sype (1970) considered the speciFc heat to be a complex quantity C(X ) = C + TC=(1 + iX ), where C and TC are the non-relaxing and relaxing parts of the speciFc heat, respectively, and X = ! is the product of the angular frequency of the temperature oscillations and the relaxation time. From this deFnition, Van den Sype (1970) calculated the relaxation in the speciFc heat assuming it to contain two parts, C and TC, the non-relaxing and relaxing parts, respectively. Designating the product of the angular frequency of the temperature oscillations and the relaxation time as X = !, he obtained |C(X )|2 = (C02 + C 2 X 2 )=(1 + X 2 ) ;
(11.12a)
tan T = X TC=(C0 + CX 2 ) ;
(11.12b)
where C0 = C + TC is the equilibrium speciFc heat measured when X 2 1, and T is the change in the phase of the temperature oscillations. Fominaya et al. (1999b) also derived the above relations. From C0 ; C, and |C(X )|, one obtains X 2 = (C02 − |C(X )|2 )=(|C(X )|2 − C 2 ) :
(11.13)
The di0erence between the speciFc heats measured at a low and a high modulation frequency thus depends on C; TC, and X . The relaxation is also observable through changes in the phase of the temperature oscillations. In the adiabatic regime, the phase shift between the oscillations ◦ of the heating power and the temperature oscillations is close to 90 . Relaxation reduces this phase shift. The phase shift in the temperature oscillations depends nonmonotonically on X (Fig. 11.11). However, measurements of the phase shift are very important because they provide an additional conFrmation of the relaxation. The relaxation time decreases with increasing
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temperature. For a given modulation frequency, the ratio of the speciFc heats measured at a low and a high frequency of the temperature oscillations, C0 = |C(X )|, Frst increases with temperature, reaches a maximum, and then falls because the relaxation time decreases. Smith (1966) carried out the Frst calorimetric measurements with various modulation frequencies on germanium whiskers. He varied the frequencies from 5.7 to 92 Hz. The ratio c=R was measured by the third-harmonic technique. It turned out that at frequencies above 10 Hz the values obtained are up to 20% higher than at lower frequencies. The author concluded that the phenomenon is a result of changes in R due to the generation of point defects in the crystal lattice. At higher frequencies, the defect concentration is unable to follow the temperature oscillations, which results in changes of R . However, an analysis of this conjecture (Smith and Holland, 1966) led to a dramatical contradiction with existing data on self-di0usion in germanium. The di0erence between the low- and high-frequency data practically did not depend on temperature. Further attempts to observe the relaxation in speciFc heat were undertaken with modulation frequencies up to 103 Hz. Measurements on platinum have shown no relaxation (Seville, 1974). Modulation measurements on gold by Skelskey and Van den Sype (1974) have shown a frequency dependence of the quantity c=R . The increase in this quantity at high frequencies is explainable if the relative vacancy contribution to the temperature derivative of resistance is larger than that to the speciFc heat. The measurements were carried out at a single temperature, 1164 K, so that the result could not be supported by the temperature dependence of the phenomenon. Measurements with various modulation frequencies were also done by Greene et al. (1972), Eno et al. (1977), and Suzuki et al. (1982). At high temperatures, high-frequency temperature oscillations were observed by means of photosensors (Kraftmakher, 1981). 11.4.2. Speci@c-heat spectroscopy of glass transitions An outstanding achievement of modulation calorimetry is speciFc-heat spectroscopy (Birge and Nagel, 1985, 1987; Birge, 1986). It was aimed at measurements of the frequency dependence of the speciFc heat of supercooled organic liquids near the glass transitions. A thin nickel Flm evaporated onto a glass substrate is immersed in the liquid and generates temperature waves in it (Fig. 11.12). The probe serves as a heater and a thermometer simultaneously. From the temperature oscillations in the probe, one obtains the e0usivity of the sample, (*c))1=2 , where *c is the speciFc heat per unit volume, and ) is the thermal conductivity of the sample. This quantity becomes complex and frequency dependent when the sample approaches the glass-transition region. The measurements employ the third-harmonic technique and are possible in a wide frequency range. Jung et al. (1992) designed an automated calorimeter that operates in the range 10 mHz–10 kHz (Fig. 11.13). SpeciFc-heat spectroscopy has become an important branch of calorimetry. It is worth remembering that all the basic components of speciFc-heat spectroscopy have been found long before. The third-harmonic method was invented by Corbino (1911) and further developed by Filippov (1960), Rosenthal (1961, 1965), and Holland (1963). Filippov (1960) was the Frst to employ a heater-thermometer probe for modulation measurements of the e0usivity of liquids. Rosenthal (1965) also used this method to sense the medium in which the probe was immersed. Jackson and Koehler (1960) and Korosto0 (1962) considered the frequency-dependent
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Fig. 11.12. Three sensors for various frequency ranges used in speciFc-heat spectroscopy (Menon, 1996). Fig. 11.13. Apparatus for speciFc-heat spectroscopy employing the third-harmonic technique (Jung et al., 1992; Moon et al., 1996).
speciFc heat in relation to point-defect equilibration. Van den Sype (1970) further elaborated this approach. A drawback of the speciFc-heat spectroscopy was that it measured only the product of the speciFc heat and thermal conductivity. Since then, the technique was improved (Dixon and Nagel, 1988; Menon, 1996). A reliable separation of the speciFc heat and thermal conductivity is possible by using two or more heaters of di0erent widths. Using the photoacoustic technique, B[uchner and Korpiun (1987) observed a frequencydependent speciFc heat of [Ca(NO)3 ]0:4 (KNO3 )0:6 near the glass transition. Korus et al. (1997) investigated glass transitions in polymers. Inada et al. (1990) and Ema and Yao (1997) described multifrequency calorimeters. Loponen et al. (1982), Berret et al. (1992), and Rajeswari and Raychaudhuri (1993) also considered time-dependent speciFc heat. Recently, Akutsu et al. (1999) observed frequency-dependent step-like anomalies in the speciFc heat of organic conductors (DMET)2 BF4 and (DMET)2 ClO4 in the range 100 –130 K. The phenomenon, typical for glass transitions, was attributed to the freezing of the intramolecular motion of the ethylene group in the DMET molecule. A similar phenomenon was observed in (-(BEDT-TTF)2 Cu[N(CN)2 ]Br, in the range 90 –120 K (Saito et al., 1999). This compound is an organic superconductor at low temperatures. Low frequencies, in the range 1–16 Hz, appeared to be suIcient to observe the relaxation. In a review of speciFc-heat spectroscopy, Birge et al. (1997) pointed out two main avenues for improvement of this technique: (i) extending the bandwidth to higher frequencies and (ii) enhancing the sensitivity. They described the situation as follows: “Unfortunately, at frequencies greater than 10 kHz an additional third-harmonic signal generated by a di0erent mechanism becomes the dominated signal. Although it has not been proved, it appears that the mechanism is due to thermally induced acoustic resonances of the heater substrate: : : We have not yet devised a scheme to defeat this spurious signal”. Menon (1996) pointed out that when the third-harmonic signal is to be measured within 1% accuracy, stray third-harmonic voltages must be held
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Fig. 11.14. Output voltages of the fundamental and of the third-harmonic frequency versus input voltage applied to both inputs of the di0erential ampliFer.
below about 20 nV. He claimed that “in principle, the high-frequency limit in the speciFc-heat spectroscopy is set only by the frequency scale at which the thermal di0usion length becomes comparable to the thickness of the heater Flm. This limit is at least two decades above the highest frequency used in this experiment, which is only 4 kHz. Above this frequency, the parasitic third-harmonic signal becomes too large to be reliably treated as a calibrated background.” As a rule, a bridge serves to balance the fundamental-frequency voltage across the sensor and to extract the third-harmonic signal. However, instead of directly measuring the output voltage of the bridge, voltages across the sensor and across another resistor of the bridge are fed to two inputs of a di0erential ampliFer. Relatively large voltages are thus fed to the ampliFer’s inputs. Jeong (1997) pointed out that the upper limit in the modulation frequency is posed by an insuIcient common-mode rejection by the di0erential preampliFer. However, this ratio governs appearance of an output voltage of the fundamental frequency. Much more important is that, due to nonlinearity, the ampliFer creates a third-harmonic voltage. This conjecture was checked by means of a lock-in ampliFer with di0erential input, PAR 124A. A sine-wave voltage from a multifunction synthesizer, HP 8904A, was fed to its inputs. At 3 kHz, the amplitude of this voltage was varied in the range 0.1–3 V. The ampliFer PAR 124A incorporates a selective ampliFer. It was in turn tuned to the fundamental and the third-harmonic frequency. The ampliFed voltage was observed by means of an oscilloscope. The second channel of the multifunction synthesizer served as the source of the 3! reference when measuring the third-harmonic signal created by the ampliFer. The results obtained (Fig. 11.14) show that the output imbalance signal at the fundamental frequency is proportional to the input signal, and the common-mode-rejection ratio amounts to 114 dB. The ampliFer generates also a third-harmonic signal. This signal nonlinearly depends on the input voltage, increasing from about 10 nV for 0:3 V at the input to about 300 nV for 3 V at the input. This third-harmonic signal should not depend on the frequency, in agreement with observations by Menon (1996). The above conjecture is thus quite realistic. Probably, the unwanted e0ect is avoidable by directly measuring the output voltage
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Table 11.9 Modulation calorimetry of glass transitionsa Item
Reference
Glycerol, propylene glycol, 0.01–6000 Hz CKN, photoacoustic technique (o-terphenyl)1−x -(o-phenylphenol)x Salol Glycerol, 10 –10 000 Hz KDP, CKN, 0.01–5000 Hz
Birge and Nagel (1985, 1987), Birge (1986) B[uchner and Korpiun (1987) Dixon and Nagel (1988) Dixon (1990) Inada et al. (1990) Jung et al. (1992), Jeong and Moon (1995), Moon et al. (1996), Jeong (1997) Cournoyer et al. (1994) Hensel et al. (1996) Wagner and Kasap (1996) Menon (1996), Birge et al. (1997) Korus et al. (1997) Beiner et al. (1998) Akutsu et al. (1999) Hu et al. (1999) Saito et al. (1999)
Polystyrene, photoacoustic technique Polymers Asx Se1−x , MDSC method Di-n-butylphthalate, 0.004 –8000 Hz Glycerol, polymers, 0.2–2000 Hz Poly(n-hexyl methacrylate), 0.2–2000 Hz (DMET)2 BF4 ; (DMET)2 ClO4 Pd 40 Ni10 Cu30 P20 , MDSC (-(BEDT-TTF)2 Cu[N(CN)2 ]Br a
CKN = [(Ca(NO)3 ]0:4 (KNO3 )0:6 ; KDP = KH2 PO4 ; BEDT-TTF = bis(ethylenedithio)tetrathiafulvalene; DMET = dimethyl(ethylenedithio):
of the bridge, which can be made several orders of magnitude smaller than the voltage across the sensor. An isolating transformer is suIcient for this purpose. However, such a transformer introduces additional phase shifts (Menon, 1996). An extension of the frequency band seems to be achievable also by using the supplementarycurrent method, instead of the third-harmonic technique. In this case, the signal to be measured has a frequency equal to the di0erence between the frequency of the temperature oscillations and that of the supplementary current. This frequency can be set in a favorable range. The supplementary current must be several times weaker than the heating current. The di0erence-frequency signal is therefore weaker than the third-harmonic signal, but it can be measured more reliably. This was shown in measurements of the temperature derivative of the speciFc heat (Kraftmakher and Tonaevskii, 1972), when the second-harmonic component in the temperature oscillations was measured in the presence of a much stronger fundamental signal. Table 11.9 lists studies of glass transitions carried out by using speciFc-heat spectroscopy. 11.4.3. Point-defect equilibration in metals In studies of point-defect equilibration in metals, the sample is subjected to such rapid temperature oscillations that the defect concentration cannot follow them. Under such conditions, the defect contribution to a given physical property is almost completely excluded. This statement relates only to properties that sense changes in the defect concentrations during the measurements: the speciFc heat, thermal expansivity, and temperature derivative of resistance. This is because the main contributions to the above properties are caused not by the presence of point defects, but by the temperature dependence of their concentration. When the defect concentration does not follow temperature oscillations and retains a mean value, these properties practically
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Fig. 11.15. Setup to observe relaxation in high-temperature speciFc heat of metals and alloys (Kraftmakher, 1981). Temperature oscillations of low and high frequencies occur in the sample simultaneously, and the ratio of corresponding speciFc heats is measured directly.
correspond to a hypothetical defect-free crystal. The only obstacle for this approach arises from short equilibration times due to the high mobility of the defects at high temperatures and numerous internal sources (sinks) for them. The relaxation is therefore observable only with high modulation frequencies. The amplitude of the temperature oscillations is inversely proportional to their frequency, and such measurements require a sensitive technique. When measuring the speciFc heat at a very high modulation frequency, the result should correspond to a defect-free crystal. At intermediate frequencies, it depends on the modulation frequency and the relaxation time. To search for relaxation phenomena in metals, a method of measuring high-temperature speciFc heat at frequencies of the order of 105 Hz has been developed (Kraftmakher, 1981). A high-frequency current slightly modulated by a low-frequency voltage heats a wire sample (Fig. 11.15). Therefore, high- and low-frequency temperature oscillations occur in the sample simultaneously. A photomultiplier detects these oscillations. The low-frequency component of its output signal proceeds to a lock-in ampliFer. The high-frequency component selected by a resonant circuit is measured using frequency conversion and lock-in detection. An auxiliary frequency converter provides the reference voltage for the lock-in detector. A plotter records a signal proportional to the di0erence between the amplitudes of the high- and low-frequency
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Fig. 11.16. Ratio of speciFc heats measured at low and high frequencies of temperature oscillations in tungsten and platinum (Kraftmakher, 1985, 1990). The high frequency was 3 × 105 and 5 × 104 Hz, respectively. Fig. 11.17. Temperature dependence of equilibration times from various measurements: 1—Au, resistivity (Seidman and BalluI, 1965); 2—Al, current noise (Celasco et al., 1976); 3—Au, positron annihilation (Schaefer and Schmid, 1989); 4—Pt, speciFc heat (Kraftmakher, 1990); 5—W, speciFc heat (Kraftmakher, 1985). The straight lines correspond to HM = 7kB TM and constant densities of sources (sinks) for vacancies.
temperature oscillations. The measurements start at temperatures where the point-defect concentration is negligible and no relaxation in speciFc heat is expected. At these temperatures, the signal is adjusted to be zero. Then the di0erence between the speciFc heats corresponding to the two frequencies is directly measured at various temperatures. The measurements were carried out on commercial 8-m tungsten wires and on vacuum incandescent lamps with 10–20 m tungsten Flaments (Kraftmakher, 1985). The highest modulation frequency was 3 × 105 Hz. The nature of the temperature dependence of the relaxation was always within the expectation (Fig. 11.16). This observation gained no recognition. For instance, Trost et al. (1986) concluded that “on the basis of the information available at present we cannot exclude with certainty that the observed e0ect is partly or even entirely due to the experimental procedure and hence not intrinsic.” In the case of platinum, the high frequency was 5 × 104 Hz (Kraftmakher, 1990). The samples were cut from a foil 10 m thick. Due to the lower melting temperature, the power heating the sample and the amplitude of the power oscillations become smaller, which results in decrease of the applicable modulation frequency. The observed relaxation also was in agreement with the nonlinear increase in the speciFc heat. As for tungsten, the scatter of the experimental points increases at temperatures where X is close to unity. This is quite expectable since only in this range does the relaxation depend strongly on X . The quenched-in resistivity and changes in the positron-annihilation parameters are certainly caused by vacancies. At the same time, the relation between vacancy formation and the nonlinear increase in speciFc heat is not commonly accepted. It seems therefore useful to compare
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results of all relaxation experiments on various metals (Fig. 11.17). The relaxation time depends on the density of sources (sinks) for vacancies. The di0erence between the relaxation times in gold obtained by measurements of the resistivity (Seidman and BalluI, 1965) and by positron annihilation (Schaefer and Schmid, 1989), which amounts to a factor of 50, is therefore quite understandable. Various densities of internal defects are probably responsible for such a di0erence in platinum samples. The migration enthalpies, HM , were considered to be proportional to the melting temperatures: HM = 7kB TM . The straight lines in the graph correspond to = B exp(HM =kB T ), where HM is the migration enthalpy, and B is a factor di0erent for various samples. Assuming a temperature-independent density of sources (sinks) for vacancies, the temperature dependence of the relaxation time is available from a single measured value. In tungsten, the relaxation times are 5 × 10−7 s at 2600 K and 2 × 10−7 s at 2700 K. The short relaxation times are consistent with the well-known fact that dislocation densities in refractory metals are much higher than those in metals such as gold or platinum. The comparison of the data is thus favorable for the conclusion that all the phenomena are of a common nature. No direct evidence exists that the phenomenon observed originates from point-defect equilibration. A simple experimental approach was proposed to check this conjecture. The relaxation should be observed during a period including a quench and consequent annealing of the sample. The main sources and sinks for vacancies are dislocations and, probably, vacancy clusters. Their density dramatically increases after a quench, and a certain time is necessary to anneal the sample at the high temperature. If, while the relaxation is observed, the heating current is interrupted and then the sample is returned to the initial temperature, the phenomenon may disappear. It will appear again after a proper anneal of the sample and recovery of its structure. The amplitude and the phase of the high-frequency temperature oscillations should not alter when the sample is quenched from a low temperature, where TC C or the relaxation time remains suIciently long (X 2 1) even after the quench. No changes are expected also at high temperatures where the modulation frequency becomes insuIcient to observe the relaxation (X 2 1). Only under very favorable conditions, when X 2 1 before the quench and X 2 1 after it, will the change in the amplitude of the high-frequency temperature oscillations reveal the true magnitude of the relaxation. Otherwise, this change corresponds to only a part of the phenomenon. Measurements with various modulation frequencies should make clear the temperature dependence of the phenomenon. Along with providing data on equilibration, this approach would conFrm the vacancy origin of the relaxation phenomenon in the speciFc heat. The directions of further investigations of point-defect equilibration are evident: (i) employment of pure and well-prepared samples to observe the relaxation over a wider temperature range; (ii) measurements on metals in which low dislocation densities are obtainable; and (iii) observations of changes in the relaxation time during a quench and subsequent anneal of the samples. 11.4.4. Further examples of frequency-dependent speci@c heat Garoche et al. (1978) measured the speciFc heat of TaS2 single crystals in the trigonal 2H phase between 0.3 and 3 K. In the superconducting state, below 0:6 K, the speciFc heat is frequency independent in the range 3–20 Hz. However, when the sample was restored to
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the normal state by applying external magnetic Feld, the speciFc heat was found to decrease with increasing frequency. The authors explained this result by a frequency-dependent nuclear contribution to the speciFc heat. The phenomenon is governed by the rate at which equilibrium is established between the nuclear spins and the lattice. Jeong et al. (1991) claimed that there is an indication that the speciFc-heat peak associated with the magnetic phase transition in gadolinium “may depend on the measuring frequencies”. This conclusion was derived from a minor di0erence in the speciFc-heat data obtained with modulation frequencies 0.24 and 316 Hz. An increase of the upper frequency would clarify the question. Saruyama (1997) observed a frequency dependence of the speciFc heat of polyethylene in the vicinity of the melting point. Minakov et al. (1999) discussed the complex heat capacity. Fominaya et al. (1999b) found a relaxation phenomenon in the speciFc heat of an Mn12 acetate crystal. The crystals are arrangements of identical magnetic clusters in an organic matrix, without exchange coupling from one cluster to another. Each cluster is composed of 12 manganese ions coupled in a ferrimagnetical conFguration to an S = 10 macrospin. A strong crystalline anisotropy lifts the 2S + 1 degeneracy of the magnetic levels in zero Feld, creating a conFguration where two levels are separated by a barrier of about 60 K. The relaxation time of the magnetization at low temperatures is therefore very long. The magnetic contribution to the speciFc heat and the relaxation time depend on temperature and applied magnetic Feld. The magnetic Feld parallel to the easy axis of magnetization was varied from −1 to 1 T. The modulation frequencies were in the range 4–20 Hz. From the measurements, the relaxation time was calculated for various temperatures and magnetic Felds. Fominaya et al. (1999a) observed similar behavior of speciFc heat in magnetic Feld in Fe8 crystals, but without any indication of a frequency dependence in the range 35–700 Hz. 12. Conclusion In conclusion, it is worth attracting attention to some points presented in the review. (1) Measurements of high-temperature speciFc heat employing reference samples, such as tungsten or platinum. When provided with blackbody models, such samples provide temperature oscillations of deFnite amplitude. These oscillations can be compared with those in other samples with a blackbody model (Section 4.2). (2) Application of direct measurements of the temperature coeIcient of the speciFc heat (Section 5.2.3) to studies of phase transitions. (3) Determinations of the speciFc heat of nonconducting materials at high temperatures. In particular, the equivalent-impedance technique or a capillary Flled with a material under study may be employed (Section 5.2.7). (4) All variants of modulation dilatometry are promising, especially for high-temperature studies (Sections 6.2– 6.7). (5) Modulation measurements of spectral absorptance (Section 7.3) are capable of extending the temperature range and enhancing the accuracy of data. (6) Direct measurements of temperature oscillations in wire samples through their thermal noise (Sections 8.1–8.3) would make it possible to more widely use such samples in all the modulation techniques.
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(7) The unusual premelting anomaly in speciFc heat (Section 11.2.4) deserves further studies, also by means of other techniques. (8) It is desirable to improve the method of observing temperature Euctuations in small samples at high temperatures (Section 11.3) and to obtain further data on isochoric speciFc heat of metals. (9) The possibility to extend the frequency band in speciFc-heat spectroscopy (Section 11.4.2) should be examined. (10) Further e0orts are necessary to clarify the origin of the relaxation phenomenon in the high-temperature speciFc heat of metals (Section 11.4.3). Acknowledgements Many thanks to the editor, Professor A.A. Maradudin, for his work for improving the review. The support of the Ministry of Science and Technology of Israel is gratefully acknowledged. References A0ortit, C., Lallement, R., 1968. Rev. Int. Hautes TempRer. et RRefract. 5, 19–26. Akhmatova, I.A., 1965. Dokl. Akad. Nauk SSSR 162, 127–129 ∗. Akhmatova, I.A., 1967. Izmer. Tekhn. N 8, 14–17. Akimov, A.I., Kraftmakher, Y.A., 1970. Phys. Stat. Sol. 42, K41–42. Akutsu, H., Saito, K., Sorai, M., 2000. Phys. Rev. B 61, 4346–4352. Akutsu, H., Saito, K., Yamamura, Y., Kikuchi, K., Nishikawa, H., Ikemoto, I., Sorai, M., 1999. J. Phys. Soc. Japan 68, 1968–1974 ∗. Akutsu, N., Ikeda, H., 1981. J. Phys. Soc. Japan 50, 2865–2871. Annino, A., Grasso, F., Musumeci, F., Triglia, A., 1984. Appl. Phys. A 35, 115–118. Anshukova, N.A., Bugoslavskiy, Y.V., Veselago, V.G., Golovashkin, A.I., Ershov, O.V., Zaytzev, I.A., Ivanenko, O.M., Minakov, A.A., Mitzen, K.V., 1989. Acta Phys. Polonica A 76, 35–40. Arpaci, E., Frohberg, M.G., 1984. Z. Metallkunde 75, 614–618. Ashman, J., Handler, P., 1969. Phys. Rev. Lett. 23, 642–644. Ausloos, M., Benhaddou, M., Cloots, R., 1994. Physica C 235=240, 1767–1768. Bachmann, R., DiSalvo, F.J., Geballe, T.H., Greene, R.L., Howard, R.E., King, C.N., Kirsch, H.C., Lee, K.N., Schwall, R.E., Thomas, H.-U., Zubeck, R.B., 1972. Rev. Sci. Instr. 43, 205–214. Baloga, J.D., Garland, C.W., 1977. Rev. Sci. Instr. 48, 105–110 ∗. Baur, H., Wunderlich, B., 1998. J. Thermal Anal. 54, 437–465. Bednarz, G., Geldart, D.J.W., White, M.A., 1993. Phys. Rev. B 47, 14247–14259. Bednarz, G., Millier, B., White, M.A., 1992. Rev. Sci. Instr. 63, 3944–3952. Beiner, M., Kahle, S., Hempel, E., Schr[oter, K., Donth, E., 1998. Europhys. Lett. 44, 321–327. Berret, J.-F., Meissner, M., Mertz, B., 1992. Z. Phys. B 87, 213–217. Bezemer, J., Jongerius, R.T., 1976. Physica C 83, 338–346. Biondi, M.A., 1954. Phys. Rev. 96, 534–535. Biondi, M.A., 1956. Phys. Rev. 102, 964–967. Birge, N.O., 1986. Phys. Rev. B 34, 1631–1642. Birge, N.O., Dixon, P.K., Menon, N., 1997. Thermochim. Acta 304=305, 51–66. Birge, N.O., Nagel, S.R., 1985. Phys. Rev. Lett. 54, 2674–2677 ∗ ∗ ∗. Birge, N.O., Nagel, S.R., 1987. Rev. Sci. Instr. 58, 1464–1470. Birmingham, J.T., Richards, P.L., Meyer, H., 1996. J. Low Temp. Phys. 103, 183–208 ∗.
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Physics Reports 356 (2002) 119–228
Two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms V.M. Shabaev Department of Physics, St. Petersburg State University, Oulianovskaya Street 1, Petrodvorets, 198504 St. Petersburg, Russia Received October 2000; editor : J: Eichler
Contents 1. Introduction 2. Energy levels of atomic systems 2.1. 2N -time Green’s function 2.2. Two-time Green’s function (TTGF) and its analytical properties 2.3. Energy shift of a single level 2.4. Perturbation theory for degenerate and quasi-degenerate levels 2.5. Practical calculations 2.6. Nuclear recoil corrections 3. Transition probabilities and cross sections of scattering processes 3.1. Photon emission by an atom 3.2. Transition probability in a one-electron atom 3.3. Radiative recombination of an electron with an atom 3.4. Radiative recombination of an electron with a high-Z hydrogen-like atom 3.5. Photon scattering by an atom 3.6. Resonance scattering: spectral line shape
121 122 123 126 131 135 139 163 167 167 171 177 182 187 191
3.7. Resonance photon scattering by a oneelectron atom 4. Numerical evaluations of QED and interelectronic-interaction corrections in heavy ions 4.1. Methods of numerical evaluations and renormalization procedure 4.2 Energy levels in heavy ions 4.3 Hyper;ne splitting and bound-electron g factor 4.4. Radiative recombination of an electron with a heavy ion 5. Conclusion Acknowledgements Appendix A. QED in the Heisenberg representation Appendix B. Singularities of the two-time Green’s function in a ;nite order of perturbation theory Appendix C. Two-time Green’s function in terms of the Fourier transform of the 2N -time Green’s function
E-mail address:
[email protected] (V.M. Shabaev). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 2 4 - 2
193 195 195 197 202 206 216 217 217 217 221
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Appendix D. Matrix elements of the two-time Green’s function between one-determinant wave functions
222
Appendix E. Double spectral representation for the two-time Green’s function describing a transition process References
223 224
Abstract The two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms is described in detail. This method provides a simple procedure for deriving formulas for the energy shift of a single level and for the energies and wave functions of degenerate and quasi-degenerate states. It also allows one to derive formulas for the transition and scattering amplitudes. Application of the method to resonance scattering processes yields a systematic theory for the spectral line shape. The practical ability of the method is demonstrated by deriving formulas for the QED and interelectronic-interaction corrections to energy levels and transition and scattering amplitudes in one-, two-, and three-electron atoms. Numerical calculations of the Lamb shift, the hyper;ne splitting, the bound-electron g factor, and c 2002 Elsevier Science B.V. the radiative recombination cross section in heavy ions are also reviewed. All rights reserved. PACS: 12.20.−m; 12.20.Ds; 31.30.Jv
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1. Introduction A great progress in experimental investigations of high-Z few-electron systems (see, e.g., [1,2]) stimulated theorists to perform accurate calculations for these systems in the framework of quantum electrodynamics (QED). The calculations of QED and interelectronic-interaction corrections in high-Z few-electron systems are conveniently divided into two stages. The ;rst stage consists in deriving formal expressions for these corrections from the ;rst principles of QED. The second comprises numerical evaluations of these expressions. The present paper is mainly focused on the ;rst stage. As to the numerical calculations, we give only a short overview of them in this paper. For more details we refer to [3–7]. Historically, the ;rst method suitable for deriving the formal expressions for the energy shift of a bound-state level was formulated by Gell-Mann and Low [8] and by Sucher [9]. This method is based on introducing an adiabatically damped factor, exp(−|t |), in the interaction Hamiltonian and expressing the energy shift in terms of the so-called adiabatic S matrix elements. Due to its simple formulation, the Gell-Mann–Low–Sucher formula for the energy shift gained wide spreading in the literature related to high-Z few-electron systems [10 –17]. However, the practical use of this method showed that it has several serious drawbacks. One of them is the very complicated derivation of the formal expressions for the so-called reducible diagrams. By “reducible diagrams” we denote those diagrams where an intermediate-state energy of the atom coincides with the reference-state energy. (This terminology is quite natural since it can be considered as an extension of the de;nitions introduced by Dyson [18] and by Bethe and Salpeter [19] to high-Z few-electron atoms.) As to irreducible diagrams, i.e. those diagrams where the intermediate-state energies diIer from the reference-state energy, the derivation of the formal expressions can easily be reduced to the “usual (=0) S-matrix” elements in each method, including the Gell-Mann–Low–Sucher method as well (see, e.g., [12,16]). Another serious drawback of the Gell-Mann–Low–Sucher method is the fact that this method requires special investigation of the renormalization procedure since the adibatic S -matrix suIers from ultraviolet divergences. The adiabatically damped factor, exp(−|t |), is non-covariant and, therefore, the ultraviolet divergences cannot be removed from S if = 0. However, from the physical point of view one may expect these divergenes to cancel each other in the expression for the energy shift. Therefore, they may be disregarded in the calculation of the energy shift for a single level. For the case of degenerate levels, however, this problem remains since we cannot expect that the standard renormalization procedure makes the secular operator ;nite in the ultraviolet limit [11,13]. In addition, we should note that at present there is no formalism based on the Gell-Mann– Low–Sucher approach which would provide a proper treatment of quasi-degenerate levels. Also no formalism in the framework of this approach was developed for calculation of the transition or scattering amplitudes up to now. The same diKculties emerge in the evolution operator method developed in Refs. [20 –24]. Attempts to solve some of these problems by modifying the Gell-Mann–Low–Sucher method were recently undertaken in [25 –27]. Another way to formulate a perturbation theory for high-Z few-electron systems consists in using Green’s functions. These functions contain the complete information about the energy levels and the transition and scattering amplitudes. In this way, the renormalization problem does not appear, since Green’s functions can be renormalized from the very beginning (see, e.g., [28]). Up to now, various versions of the Green’s function formalism were developed which diIer
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from each other by the methods of extracting the physical information from Green’s functions, i.e. the energy levels and the transition and scattering amplitudes. One of these methods was worked out in [29 –33]. It was successfully employed in many practical calculations [34 – 46]. Since one of the key elements of this method consists in using two-time Green’s functions, in what follows, we will call it the two-time Green’s function (TTGF) method. This method, which provides a solution of all the problems appearing in the other formalisms indicated above, will be considered in detail in the present work. As to other versions of the Green’s function method [13,17,47–55], a detailed discussion of them would be beyond the scope of the present paper. We note only that some of these methods are also based on employing two-time Green’s functions but yield other forms of perturbation theory. In Refs. [13,48–51], the two-time Green’s functions were used for constructing quasi-potential equations for high-Z few-electron systems. This corresponds to the perturbation theory in the Brillouin–Wigner form while the method of Refs. [29 –33] yields the perturbation theory in the Rayleigh–SchrNodinger form. Various versions of the Bethe–Salpeter equation derived from the 2N -time Green’s function formalism for high-Z few-electron systems can be found in [13,52]. In [53,54] the perturbation theory in the Rayleigh–SchrNodinger form is constructed for the case of a one-electron system where the problem of relative time coordinates for the electrons does not occur. The relativistic unit system (˝ = c = 1) and the Heaviside charge unit ( = e2 =4 ; e ¡ 0) are used in the paper. 2. Energy levels of atomic systems In this section we formulate the perturbation theory for the calculation of the energy levels in high-Z few-electron atoms. In these systems the number of electrons denoted by N is much smaller than the nuclear charge number Z. It follows that the interaction of the electrons with each other and with the quantized electromagnetic ;eld is much smaller (by factors 1=Z and , respectively) than the interaction of the electrons with the Coulomb ;eld of the nucleus. Therefore, it is natural to assume that in zeroth approximation the electrons interact only with the Coulomb ;eld of the nucleus and obey the Dirac equation (−i · B + m + VC (x)) n (x) = n
n (x)
:
(1)
The interaction of the electrons with each other and with the quantized electromagnetic ;eld is accounted for by perturbation theory. In this way we obtain quantum electrodynamics in the Furry picture. It should be noted that we could start also with the Dirac equation with an eIective potential VeI (x) which approximately describes the interaction with the other electrons. In this case the interaction with the potential V (x) = VC (x) − VeI (x) must be accounted for perturbatively. Using the eIective potential provides an extension of the theory to many-electron atoms where, for instance, a local version of the Hartree–Fock potential can be used as VeI (x). However, for simplicity, in what follows, we will assume that in zeroth approximation the electrons interact only with the Coulomb ;eld of the nucleus. In the present paper we will mainly consider the perturbation theory with the standard QED vacuum. The transition to the formalism in which closed shells are regarded as belonging
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to the vacuum is realized by replacing i0 with −i0 in the electron propagator denominators corresponding to the closed shells. Before formulating the perturbation theory for calculations of the interelectronic interaction and radiative corrections to the energy levels, we consider standard equations of the Green’s function approach in quantum electrodynamics. 2.1. 2N -time Green’s function It can be shown that the complete information about the energy levels of an N -electron atom is contained in the Green’s function de;ned as G(x1 ; : : : ; xN ; x1 ; : : : ; xN ) = 0|T (x1 ) · · · (xN ) P (xN ) · · · P (x1 )|0 ; (2) where (x) is the electron–positron ;eld operator in the Heisenberg representation, P (x) = † 0 , and T is the time-ordered product operator. The basic equations of quantum electrodynamics in the Heisenberg representation are summarized in Appendix A. Eq. (2) presents a standard de;nition of the 2N -time Green’s function which is a fundamental object of quantum electrodynamics. It can be shown (see, e.g., [28,56]) that in the interaction representation the Green’s function is given by G(x1 ; : : : ; xN ; x1 ; : : : ; xN ) =
0|T
=
in (x1 ) · · ·
∞ (−i)m m=0
m!
P (x1 ) exp{−i d 4 z HI (z)}|0 in 0|T exp{−i d 4 z HI (z)}|0
P in (xN ) in (xN ) · · ·
d 4 y1 · · · d 4 ym 0|T
in (x1 ) · · ·
P in (xN ) in (xN ) · · ·
P (x1 ) in
(3)
∞ −1 (−i)l d 4 z1 · · · d 4 zl 0|T HI (z1 ) · · · HI (zl )|0 ; × HI (y1 ) · · · HI (ym )|0 l=0
where
l!
(4)
e m P (5) [ (x); in (x)] in (x)]Ain (x) − 2 2 in is the interaction Hamiltonian. The commutators in Eq. (5) refer to operators only. The ;rst term in (5) describes the interaction of the electron–positron ;eld with the quantized electromagnetic ;eld and the second one is the mass renormalization counterterm. We consider here that the interaction of the electrons with the Coulomb ;eld of the nucleus is included in the unperturbed Hamiltonian, i.e. the Furry picture. However, there is also an alternative method to get the Furry picture. In that method the interaction with the Coulomb ;eld of the nucleus is included in the interaction Hamiltonion and the Furry picture is obtained by summing in;nite sequences of Feynman diagrams describing the interaction of the electrons with the Coulomb potential. As a result of this summation, the free-electron propagators are replaced by bound-electron propagators. This method is very convenient for studying processes involving continuum-electron states. It will be used in the section concerning the radiative recombination process. HI (x) = [ P in (x) ;
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The Green’s function G is constructed by perturbation theory according to Eq. (4). This is carried out with the aid of the Wick theorem (see, e.g., [28]). According to this theorem the time-ordered product of ;eld operators is equal to the sum of normal-ordered products with all possible contractions between the operators T (ABCD · · ·) = N (ABCD · · ·) + N (Aa Ba CD · · ·) + N (Aa BC a D · · ·) + all possible contractions ;
(6)
where N is the normal-ordered product operator and the superscripts denote the contraction between the corresponding operators. The contraction between neighbouring operators is de;ned by Aa Ba = T (AB) − N (AB) = 0|T (AB)|0 :
(7)
If the contracted operators are boson operators, they can be put one next to another. If the contracted operators are fermion operators, they also can be put one next to another but in this case the expression must be multiplied with the parity of the permutation of the fermion operators. Since in the Green’s function the vacuum expectation value is calculated, only the term with all operators contracted remains on the right-hand side of Eq. (6). In contrast to the free-electron QED, in the Furry picture the time-ordered product of two fermion operators must be de;ned also for the equal-time case to obtain the correct vacuum polarization terms. As was noticed in [57], the de;nition T [A(t)B(t)] = 12 A(t)B(t) − 12 B(t)A(t)
(8)
provides the following simple rule for dealing with the interaction operator. It can be written as HI (x) = e P in (x) in (x)A in (x) − m P in (x) in (x) (9) and then the Wick theorem is applied with contractions between all operators, including equaltime operators. We note that the problem of the de;nition of the time-ordered product of fermion operators at equal times does not appear at all if the alternative method for obtaining the Furry picture discussed above is employed. The contractions between the electron–positron ;elds and between the photon ;elds lead to the following propagators: ∞ n (x) P (y) i n 0|T in (x) P in (y)|0 = d! exp[ − i!(x0 − y0 )] (10) 2 −∞ ! − (1 − i 0) n n and 0|TA in (x)A$in (y)|0
= −ig
$
d 4 k exp[ − ik · (x − y)] : (2 )4 k 2 + i0
(11)
Here the Feynman gauge is considered. In Eq. (10) the index n runs over all bound and continuum states. The denominator in Eq. (3) describes unobservable vacuum–vacuum transitions and, as can be shown (see, e.g., [28]), it cancels disconnected vacuum–vacuum subdiagrams from the numerator. Therefore, we can simply omit all diagrams containing disconnected vacuum–vacuum subdiagrams in the numerator and replace the denominator by 1.
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In practical calculations of the Green’s function it is convenient to work with the Fourier transform with respect to time variables, G((p10 ; x1 ); : : : ; (pN0 ; xN ); (p10 ; x1 ); : : : ; (pN0 ; xN )) ∞ −2N = (2 ) d x10 · · · d xN0 d x10 · · · d xN0 −∞
×exp(ip10 x10 + · · · + ipN0 xN0 − ip10 x10 − · · · − ipN0 xN0 )G(x1 ; : : : ; xN ; x1 ; : : : ; xN ) :
(12)
For the Green’s function G((p10 ; x1 ); : : : ; (pN0 ; xN ); (p10 ; x1 ); : : : ; (pN0 ; xN )), the following Feynman rules can be derived: (1) External electron line: i S(!; x; y) ; 2 where n (x) P (y) n ; (13) S(!; x; y) = ! − (1 − i0) n n n (x)
are solutions of the Dirac equation (1). (2) Internal electron line: ∞ i d ! S(!; x; y) : 2 −∞ (3) Disconnected electron line: i S(!; x; y)'(! − ! ) : 2 (4) Internal photon line: ∞ i d ! D() (!; x − y) ; 2 −∞
where, for zero photon mass, D() (!; x − y) is given by d k exp(ik · (x − y)) D() (!; x − y) = −g() (2 )3 !2 − k2 + i0 in the Feynman gauge and by 1 ; Di0 = D0i = 0 (i = 1; 2; 3) ; D00 (!; x − y) = 4 |x − y| d k exp(ik · (x − y)) ki kl Dil (!; x − y) = ' − (i; l = 1; 2; 3) il (2 )3 !2 − k2 + i0 k2
(14)
(15) (16)
in the Coulomb gauge. (In this work we assume that the Coulomb gauge is used only for diagrams which do not involve a renormalization procedure. The renormalization in the Coulomb gauge is discussed in Refs. [58,59].)
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(5) Vertex:
(
−2 ie '(!1 − !2 − !3 )
dx :
(6) The mass counterterm:
2 i'(! − ! )'m
dx :
(7) Symmetry factor (−1)P , where P is the parity of the permutation of the ;nal electron coordinates with respect to the initial ones. (8) Factor (−1) for every closed electron loop. (9) If, in addition, an external potential V (x) is present, an additional vertex appears,
0
−2 i (! − ! )
d x V (x) :
In principle, the Green’s function G contains the complete information about the energy levels of the atomic system. This can be shown by deriving the spectral representation for G. However, it is a hard task to extract this information directly from G since it depends on 2(N − 1) relative times (energies) in the time (energy) representation. As we will see in the next section, the two-time Green’s function de;ned as ˜ ; t) ≡ G(t1 = t2 = · · · = tN ≡ t ; t1 = t2 = · · · = tN ≡ t) G(t (17) also contains the complete information about the energy levels, and it is a much simpler task ˜ to extract the energy levels from G. 2.2. Two-time Green’s function (TTGF) and its analytical properties Let us introduce the Fourier transform of the two-time Green’s function by G(E; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E − E ) ∞ 1 1 d x0 d x0 exp(iE x0 − iEx0 ) = 2 i N ! −∞ ×0|T (x0 ; x1 ) · · · (x0 ; xN ) P (x0 ; xN ) · · · P (x0 ; x1 )|0 ;
(18)
where, as in (2), the Heisenberg representation for the electron–positron ;eld operators is used. De;ned by Eq. (18) for real E, the Green’s function G can be continued analytically to the complex E-plane. Analytical properties of this type of Green’s functions in the complex E-plane
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were studied in various ;elds of physics (see, e.g., [60 – 62]). In quantum ;eld theory they were considered in detail by Logunov and Tavkhelidze [63] (see also [64]), where the two-time Green’s function was employed for constructing a quasi-potential equation. To study the analytical properties of the two-time Green’s function we derive the spectral representation for G. Using the time-shift transformation rule for the Heisenberg operators (see Appendix A) (x0 ; x) = exp(iHx0 ) (0; x) exp(−iHx0 ) and the equations H |n = En |n;
(19)
|nn| = I ;
(20)
n
where H is the Hamiltonian of the system in the Heisenberg representation, we ;nd G(E; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E − E ) ∞ 1 1 d x0 d x0 exp(iE x0 − iEx0 ) = 2 i N ! −∞ × .(x0 − x0 ) exp[i(E0 − En )(x0 − x0 )]0| (0; x1 ) · · · (0; xN )|n n
2 ×n| P (0; xN ) · · · P (0; x1 )|0 + (−1)N .(x0 − x0 ) exp[i(E0 − En )(x0 − x0 )] n
×0| P (0; xN ) · · · P (0; x1 )|nn| (0; x1 ) · · · (0; xN )|0
:
(21)
Assuming, for simplicity, E0 = 0 (it corresponds to choosing the vacuum energy as the origin of reference) and taking into account that ∞ d x0 d x0 .(x0 − x0 ) exp[ − iEn (x0 − x0 )] exp[i(E x0 − Ex0 )] −∞
= 2 '(E − E)
∞
−∞
i
E − En + i0
;
(22)
d x0 d x0 .(x0 − x0 ) exp[ − iEn (x0 − x0 )] exp[i(E x0 − Ex0 )]
= − 2 '(E − E)
i
E + En − i0
;
(23)
we obtain G(E) =
n
/n /P n 0n 0P n − (−1)N ; E − En + i0 E + En − i0 n
(24)
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where the variables x1 ; : : : ; xN ; x1 ; : : : ; xN are implicit and 1 /n (x1 ; : : : ; xN ) = √ 0| (0; x1 ) · · · (0; xN )|n ; N!
(25)
1 0n (x1 ; : : : ; xN ) = √ n| (0; x1 ) · · · (0; xN )|0 : (26) N! In Eq. (24) the summation runs over all bound and continuum states of the system of the interacting ;elds. Let us introduce the functions A(E; x1 ; : : : ; xN ; x1 ; : : : ; xN ) = '(E − En )/n (x1 ; : : : ; xN )/P n (x1 ; : : : ; xN ) ; (27) n
B(E; x1 ; : : : ; xN ; x1 ; : : : ; xN ) =
n
'(E − En )0n (x1 ; : : : ; xN )0P n (x1 ; : : : ; xN ) :
(28)
These functions satisfy the conditions ∞ 1 d E A(E; x1 ; : : : ; xN ; x1 ; : : : ; xN ) = 0| (0; x1 ) · · · (0; xN ) P (0; xN ) · · · P (0; x1 )|0 ; N! −∞ (29) ∞ 1 d E B(E; x1 ; : : : ; xN ; x1 ; : : : ; xN ) = 0| P (0; xN ) · · · P (0; x1 ) (0; x1 ) · · · (0; xN )|0 : N ! −∞ (30) In terms of these functions, Eq. (24) is ∞ ∞ A(E ) B(E ) N G(E) = dE − ( − 1) d E ; E − E + i0 E + E − i0 0 0
(31)
where we have omitted the variables x1 ; : : : ; xN ; x1 ; : : : ; xN and have taken into account that A(E ) = B(E ) = 0 for E ¡ 0 since En ¿ 0. In fact, due to charge conservation, only states with an electric charge of eN contribute to A in the sum over n in the right-hand side of Eq. (27) and only states with an electric charge of −eN contribute to B in the sum over n in the right-hand side of Eq. (28). This can easily be shown by using the following commutation relations: [Q; (x)] = −e (x); [Q; P (x)] = e P (x) ; (32) where Q is the charge operator in the Heisenberg representation. Therefore, Eq. (31) can be written as ∞ ∞ A(E ) B(E ) N G(E) = dE − ( − 1) d E ; (33) (+) (− ) E − E + i0 E + E − i0 Emin Emin (+) (−) is the minimal energy of states with electric charge eN and Emin is the minimal where Emin energy of states with electric charge −eN . So far we considered G(E) for real E. Eq. (33) shows that the Green’s function G(E) is the sum of Cauchy-type integrals. Using the fact that
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129
Fig. 1. Singularities of the two-time Green’s function in the complex E-plane. Fig. 2. Singularities of the two-time Green’s function in the bound state region, if the interaction between the electron–positron ;eld and the electromagnetic ;eld is switched oI.
the integrals E∞(+) d E A(E) and E∞(−) d E B(E) converge (see Eqs. (29) and (30)), one can show min min with the help of standard mathematical methods that the equation ∞ ∞ A(E ) N B(E ) G(E) = dE − ( − 1) d E (34) (+) (−) E − E E + E Emin Emin (−) de;nes an analytical function of E in the complex E plane with the cuts (−∞; Emin ] and (+) [Emin ; ∞) (see Fig. 1). This equation provides the analytical continuation of the Green’s function to the complex E plane. According to (33), to get the Green’s function for real E we have to approach the right-hand cut from the upper half-plane and the left-hand cut from the lower half-plane. In what follows, we will be interested in bound states of the system. According to Eqs. (24) – (34), the bound states correspond to the poles of the function G(E) on the right-hand real semiaxis. If the interaction between the electron–positron ;eld and the electromagnetic ;eld is switched oI, the poles corresponding to bound states are isolated (see Fig. 2). Switching on the interaction between the ;elds transforms the isolated poles into branch points. This is caused by the fact that due to zero photon mass the bound states are no longer isolated points of the spectrum. Disregarding the instability of excited states, the singularities of the Green’s function G(E) are shown in Fig. 3. The poles corresponding to the bound states lie on the upper boundary of the cut starting from the pole corresponding to the ground state. It is natural to assume that G(E) can be continued analytically under the cut, to the second sheet of the Riemann surface. As a result of this continuation the singularities of G(E) can be turned down as displayed in Fig. 4. In fact, due to instability of excited states the energies of these states have small imaginary components and, therefore, the related poles lie slightly below the right-hand real semiaxis (Fig. 5). However, in calculations of the energy levels and the transition and scattering amplitudes of non-resonance processes we will neglect the instability of the excited states and, therefore, will assume that the poles lie on the real axis. The imaginary parts of the energies will be taken into account when we will consider the resonance scattering processes. To formulate the perturbation theory for calculations of the energy levels and the transition and scattering amplitudes we will need to isolate the poles corresponding to the bound states
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Fig. 3. Singularities of the two-time Green’s function in the bound state region, disregarding the instability of excited states. Fig. 4. Singularities of the two-time Green’s function in the bound state region if the cuts are turned down, to the second sheet of the Riemann surface. The instability of excited states is disregarded.
Fig. 5. Singularities of the two-time Green’s function in the bound state region if the cuts are turned down, to the second sheet of the Riemann surface. The instability of excited states is taken into account. Fig. 6. Singularities of the two-time Green’s function in the bound state region for a non-zero photon mass, including one- and two-photon spectra, if the cuts are turned down, to the second sheet of the Riemann surface. The instability of excited states is disregarded.
from the related cuts. It can be done by introducing a non-zero photon mass which is generally assumed to be larger than the energy shift (or the energy splitting) of the level (levels) under consideration and much smaller than the distance to other levels. The singularities of G(E) with non-zero photon mass, including one- and two-photon spectra, are shown in Fig. 6. As one can see from this ;gure, introducing the photon mass makes the poles corresponding to the bound states to be isolated. In every ;nite order of perturbation theory the singularities of the Green’s function G(E) in the complex E-plane are de;ned by the unperturbed Hamiltonian. In quantum mechanics this fact easily follows from the expansion of the Green’s function (E − H )−1 = (E − H0 − V )−1 in powers of the perturbation potential V −1
(E − H )
=
∞
(E − H0 )−1 [V (E − H0 )−1 ]n :
(35)
n=0
As one can see from this equation, to nth order of perturbation theory the Green’s function has poles of all orders till n + 1 at the unperturbed positions of the bound state energies. This
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fact remains also valid in quantum electrodynamics for G(E) de;ned above. It can easily be checked for every speci;c diagram in ;rst and second order in . A general proof for an arbitrary diagram is given in Appendix B. 2.3. Energy shift of a single level In this section we are interested in the energy shift TEa = Ea − Ea(0) of a single isolated level a of an N -electron atom due to the perturbative interaction. The unperturbed energy Ea(0) is equal to the sum of the one-electron Dirac–Coulomb energies Ea(0) = a1 + · · · + aN ;
(36)
which are de;ned by the Dirac equation (1). In the simplest case the unperturbed wave function ua (x1 ; : : : ; xN ) is a one-determinant function 1 ua (x1 ; : : : ; xN ) = √ (−1)P Pa1 (x1 ) · · · PaN (xN ) ; (37) N! P where n are the one-electron Dirac wave functions de;ned by Eq. (1) and P is the permutation operator. In the general case the unperturbed wave function is a linear combination of the one-determinant functions 1 ua (x1 ; : : : ; xN ) = Cab √ (−1)P Pb1 (x1 ) · · · PbN (xN ) : (38) N! P b We introduce the Green’s function gaa (E) by gaa (E) = ua |G(E)01 · · · 0N |ua ≡ d x1 · · · d xN d x1 · · · d xN ua† (x1 ; : : : ; xN ) ×G(E; x1 ; : : : ; xN ; x1 ; : : : ; xN )01 · · · 0N ua (x1 ; : : : ; xN ) :
(39)
From the spectral representation for G(E) (see Eqs. (24) – (34)) we have gaa (E) = where Aa =
1 N!
Aa + terms that are regular at E ∼ Ea ; E − Ea
× a|
(40)
d x1 · · · d xN d x1 · · · d xN ua† (x1 ; : : : ; xN )0| (0; x1 ) · · · (0; xN )|a †
(0; xN ) · · ·
†
(0; x1 )|0ua (x1 ; : : : ; xN ) :
(41)
We assume here that a non-zero photon mass is introduced to isolate the pole corresponding to the bound state a from the related cut. We consider that the photon mass is larger than the energy shift under consideration and much smaller than the distance to other levels. To generate the perturbation series for Ea it is convenient to use a contour integral formalism developed ;rst in operator theory by SzNokefalvi-Nagy and Kato [65 –70]. Choosing a contour 4 in the
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Fig. 7. The contour 4 surrounds the pole corresponding to the level under consideration and keeps outside all other singularities. For simplicity, only one- and two-photon spectra are displayed.
complex E-plane in a way that it surrounds the pole corresponding to the level a and keeps outside all other singularities (see Fig. 7), we have 1 d E Egaa (E) = Ea Aa ; (42) 2 i 4 1 d E gaa (E) = Aa : (43) 2 i 4 Here we have assumed that the contour 4 is oriented anticlockwise. Dividing Eq. (42) by (43), we obtain 1 d E Egaa (E) 2 i 4 Ea = : (44) 1 d E gaa (E) 2 i 4 It is convenient to transform Eq. (44) to a form that directly yields the energy shift TEa = Ea − Ea(0) . In zeroth order, substituting the operators bn n (x) + d†n n (x) ; (45) in (0; x) = n ¿0
P (0; x) = in
n ¡0
b†n P n (x) +
n ¿0
dn P n (x)
(46)
n ¡0
into Eqs. (25) and (26) instead of (0; x) and P (0; x), respectively, and considering the states |n in (25) and (26) as unperturbed states in the Fock space, from Eqs. (24) – (26) and (39) we ;nd 1 (0) gaa = : (47) E − Ea(0) This equation can also be derived using the Feynman rules for G (see Section 2.5.1). Denoting (0) , from (44) we obtain the desired formula [29] Tgaa = gaa − gaa 1 (0) d E (E − Ea )Tgaa (E) 2 i 4 TEa = : (48) 1 1+ d E Tgaa (E) 2 i 4
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133
Fig. 8. Singularities of the photon propagator in the complex ! plane for a non-zero photon mass .
The Green’s function Tgaa (E) is constructed by perturbation theory (1) (2) Tgaa (E) = Tgaa (E) + Tgaa (E) + · · · ;
(49)
where the superscript denotes the order in . If we represent the energy shift as a series in TEa = TEa(1) + TEa(2) + · · · ; formula (48) yields 1 (1) (1) TEa = d ETE Tgaa (E) ; 2 i 4 TEa(2)
1 (2) = d E TE Tgaa (E) 2 i 4 1 1 (1) (1) − d E TE Tgaa (E) d E Tgaa (E) ; 2 i 4 2 i 4
(50)
(51)
(52)
where TE ≡ E − Ea(0) . Deriving Eqs. (44) and (48) we have assumed that a non-zero photon mass is introduced. This allows taking all the cuts outside the contour 4 as well as regularizing the infrared singularities of individual diagrams. In the Feynman gauge, the photon propagator with non-zero photon mass is d k exp(ik · (x − y)) D() (!; x − y) = −g() (53) (2 )3 !2 − k2 − 2 + i0 or, after integration, D() (!; x − y) = g()
exp(i
!2 − 2 + i0|x − y|) ; 4 |x − y|
(54)
where Im !2 − 2 + i0 ¿ 0. D() (!; x − y) is an analytical function of ! in the complex ! plane with cuts beginning at the points ! = − + i0 and − i0 (Fig. 8). The related expressions for the photon propagator with non-zero photon mass in other covariant gauges are presented, e.g., in [28].
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As was noted in the previous subsection, the singularities of the two-time Green’s function in the complex E-plane are de;ned by the unperturbed Hamiltonian if it is constructed by perturbation theory. In particular, it means that in nth order of perturbation theory gaa (E) has poles of all orders till n + 1 at the position of the unperturbed energy level under consideration. Therefore, in calculations by perturbation theory it is suKcient to consider the photon mass as a very small parameter which provides a separation of the pole from the related cut. At the end of the calculations after taking into account a whole gauge invariant set of Feynman diagrams we can put → 0. The possibility of taking the limit → 0 follows, in particular, from the fact that the contour 4 can be shrunk continuously to the point E = Ea(0) (see Fig. 7). Generally speaking, the energy shift of an excited level derived by formula (48) contains an imaginary component which is caused by its instability. This component de;nes the width of the spectral line in the Lorentz approximation (see Sections 3.6 and 3.7 for details). For practical calculations it is convenient to express the Green’s function gaa (E) in terms of the Fourier transform of the 2N -time Green’s function de;ned by Eq. (12). By using the identity 1 ∞ d x exp(i!x) = '(!) ; (55) 2 −∞ one easily ;nds (see Appendix C) ∞ 2 1 dp0 · · · dpN0 dp10 · · · dpN0 gaa (E)'(E − E ) = i N ! −∞ 1 ×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) ×ua |G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 )01 : : : 0N |ua ;
(56)
where ua |G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 )01 : : : 0N |ua ≡ d x1 · · · d xN d x1 · · · d xN ua (x1 ; : : : ; xN ) ×G((p10 ; x1 ); : : : ; (pN0 ; xN ); (p10 ; x1 ); : : : ; (pN0 ; xN )) ×01 : : : 0N ua (x1 ; : : : ; xN ) :
(57)
According to Eq. (38) the calculation of the matrix elements in (56) is reduced to the calculation of the matrix elements between the one-determinant wave functions 1 ui = √ (−1)P Pi1 (x1 ) · · · PiN (xN ) ; (58) N! P 1 (−1)P uk = √ N! P
Pk1 (x1 ) · · ·
PkN (xN )
:
(59)
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To simplify the summation procedure over the permutations in (56) which arise from the wave functions as well as from the Green’s function G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 ), it is convenient to transform Eq. (56) in the following way. Denoting GP = G01 : : : 0N , we can write 0 0 0 0 P G((p 1 ; 71 ); : : : ; (pN ; 7N ); (p1 ; 71 ); : : : ; (pN ; 7N )) 0 0 0 0 ˆ = (−1)P G((p P1 ; 7P1 ); : : : ; (pPN ; 7PN ); (p1 ; 71 ); : : : ; (pN ; 7N )) ;
(60)
P
where 7 ≡ (x; ) and is the bispinor index ( = 1– 4). Substituting (60) in (56) and using the symmetry of Gˆ with respect to the electron coordinates, for gik (E) = ui |G(E)01 · · · 0N |uk one can obtain (see Appendix D) ∞ 2 P ∗ ∗ gik (E)'(E − E ) = (−1) Pi1 (71 ) : : : PiN (7N ) dp10 : : : dpN0 dp10 : : : dpN0 i −∞ P ×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) 0 0 0 0 ˆ ×G((p 1 ; 71 ); : : : ; (pN ; 7N ); (p1 ; 71 ); : : : ; (pN ; 7N ))
×
k1 (71 ) : : :
kN (7N )
;
(61)
where repeated variables {7} imply integration (the integration over x and the summation over ). In practical calculations by perturbation theory, formula (61) must be only employed for symmetric sets of Feynman diagrams since the symmetry property was used in its derivation. 2.4. Perturbation theory for degenerate and quasi-degenerate levels In this section we are interested in the atomic levels with energies E1 ; : : : ; Es arising from unperturbed degenerate or quasi-degenerate levels with energies E1(0) ; : : : ; Es(0) . As usual, we assume that the energy shifts of the levels under consideration or their splitting caused by the interaction are much smaller than the distance to other levels. The unperturbed eigenstates form an s-dimensional subspace 9. We denote the projector on 9 by P (0) =
s k=1
Pk(0) =
s k=1
uk uk† ;
(62)
{uk }sk=1
where are the unperturbed wave functions which, in a general case, are linear combinations of one-determinant functions (see Eq. (38)). We project the Green’s function G(E) on the subspace 9 g(E) = P (0) G(E)01 : : : 0N P (0) ;
(63)
where, as in (39), the integration over the electron coordinates is implicit. As in the case of a single level, to isolate the poles of g(E) corresponding to the bound states under consideration,
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Fig. 9. The contour 4 surrounds the poles corresponding to the quasi-degenerate levels under consideration and keeps outside all other singularities. For simplicity, only one-photon spectra are displayed.
we introduce a non-zero photon mass . We assume that the photon mass is larger than the energy distance between the levels under consideration and much smaller than the distance to other levels. In this case we can choose a contour 4 in the complex E-plane in a way that it surrounds all the poles corresponding to the considered states (E (1) ; : : : ; E (s) ) and keeps outside all other singularities, including the cuts starting from the lower-lying bound states (see Fig. 9). In addition, if we neglect the instability of the states under consideration, the spectral representation (see Eqs. (24) – (34)) gives g(E) =
s ’k ’†k + terms that are regular inside of 4 ; E − E (k)
(64)
k=1
where ’k = P (0) /k ;
’†k = /k† P (0) :
(65)
As in the case of a single level, in zeroth approximation one easily ;nds g(0) (E) =
s
Pk(0)
(0) k=1 E − Ek
:
We introduce the operators K and P by 1 K≡ d E Eg(E) ; 2 i 4 1 P≡ d E g(E) : 2 i 4
(66)
(67) (68)
Using Eq. (64), we obtain K=
s i=1
P=
s i=1
E (i) ’i ’†i ;
(69)
’i ’†i :
(70)
We note here that, generally speaking, the operator P is not a projector (in particular, P 2 = P). If the perturbation goes to zero, the vectors {’i }si=1 approach the correct linearly independent
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combinations of the vectors {ui }si=1 . Therefore, it is natural to assume that the vectors {’i }si=1 are also linearly independent. It follows that one can ;nd such vectors {vi }si=1 that ’†i vk = 'ik :
(71)
Indeed, let ’i =
s
aij uj ;
vk =
j=1
s
xkl ul :
(72)
l=1
The biorthogonality condition (71) gives s
aij xkj = 'ik :
(73)
j=1
Since the determinant of the matrix {aij } is non-vanishing due to the linear independence of {’i }si=1 , system (73) has a unique solution for any ;xed k = 1; : : : ; s. From (69) – (71) we have s
Pvk =
i=1
Kvk =
’i ’†i vk = ’k ;
s i=1
E (i) ’i ’†i vk = E (k) ’k :
(74)
(75)
Hence we obtain the equation for vk , E (k) [29] Kvk = E (k) Pvk :
(76)
According to (71) the vectors vk are normalized by the condition vk† Pvk = 'k k :
(77)
The solution of Eq. (76) yields an equation for the atomic energy levels det (K − EP) = 0 :
(78)
The generalized eigenvalue problem (76) with the normalization condition (77) can be transformed by the substitution k = P 1=2 vk to the ordinary eigenvalue problem (“SchrNodinger-like equation”) [33] H
k
= E (k)
k
(79)
with the ordinary normalization condition † k k
= 'kk ;
(80)
where H ≡ P −1=2 (K)P −1=2 . The energy levels are determined from the equation det(H − E) = 0 :
(81)
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Generally speaking, the energies determined by this equation contain imaginary components which are due to the instability of excited states. In the case when the imaginary components are much smaller than the energy distance between the levels (or the levels have diIerent quantum numbers), they de;ne the widths of the spectral lines in the Lorentz approximation. In the opposite case, when the imaginary components are comparable with the energy distance between the levels which have the same quantum numbers, the spectral line shape depends on the process of the formation of the states under consideration even in the resonance approximation (see Sections 3.6 and 3.7 for details). In what follows, calculating the energy levels we neglect the instability of excited states and assume H ≡ (H + H † )=2 in Eqs. (79) and (81). The operators K and P are constructed by formulas (67) and (68) using perturbation theory K = K (0) + K (1) + K (2) + · · · ;
(82)
P = P (0) + P (1) + P (2) + · · · ;
(83)
where the superscript denotes the order in . The operator H is H = H (0) + H (1) + H (2) + · · · ;
(84)
H (0) = K (0) ;
(85)
H (1) = K (1) − 12 P (1) K (0) − 12 K (0) P (1) ;
(86)
where
H (2) = K (2) − 12 P (2) K (0) − 12 K (0) P (2) − 12 P (1) K (1) − 12 K (1) P (1) + 38 P (1) P (1) K (0) + 38 K (0) P (1) P (1) + 14 P (1) K (0) P (1) :
(87)
It is evident that in zeroth order Kik(0) = Ei(0) 'ik ;
(88)
Pik(0) = 'ik ;
(89)
Hik(0) = Ei(0) 'ik :
(90)
To derive Eqs. (76) – (79) we have introduced a non-zero photon mass which was assumed to be larger than the energy distance between the levels under consideration and much smaller than the distance to other levels. At the end of the calculations after taking into account a whole gauge invariant set of Feynman diagrams, we can put → 0. The possibility of taking this limit in the case of quasi-degenerate states follows from the fact that the cuts can be drawn to the related poles by a deformation of the contour 4 as shown in Fig. 10.
V.M. Shabaev / Physics Reports 356 (2002) 119–228
139
Fig. 10. A deformation of the contour 4 that allows drawing the cuts to the related poles in the case of quasi-degenerate states when → 0. For simplicity, only one-photon spectra are displayed.
As in the case of a single level, for practical calculations we express the Green’s function g(E) in terms of the Fourier transform of the 2N -time Green’s function ∞ 2 1 g(E)'(E − E ) = dp0 : : : dpN0 i N ! −∞ 1 ×dp10 : : : dpN0 '(E − p10 · · · − pN0 )'(E − p10 · · · − pN0 ) ×P (0) G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 )01 : : : 0N P (0) ;
(91)
where G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 ) is de;ned by Eq. (12). 2.5. Practical calculations In this section we consider some practical applications of the method in the lowest orders of perturbation theory. In what follows, we will use the notation I (!) = e2 ( ) D() (!) ;
(92)
where ( ≡ 0 ( = (1; ). In addition we will employ the following symmetry properties of the photon propagator: I (!) = I (−!);
I (!) = −I (−!) ;
(93)
which, in particular, are valid in the Feynman and Coulomb gauges. Here I (!) ≡ d I (!)= d !. 2.5.1. Zeroth order approximation Let us derive ;rst formula (47) using the Feynman rules for G. According to Eqs. (56) – (61) we have 2 gaa (E)'(E − E ) = (−1)P d x1 · · · d xN d x1 · · · d xN i P ∞ † † × Pa1 (x1 ) · · · PaN (xN ) dp10 · · · dpN0 dp10 · · · dpN0 −∞
×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 )
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×
i i S1 (p10 ; x1 ; x1 )'(p10 − p10 ) · · · SN (pN0 ; xN ; xN )'(pN0 − pN0 ) 2 2
×01 · · · 0N
2 = i
∞
−∞
a1 (x1 ) · · ·
aN (xN )
(94)
dp10 · · · dpN0 dp10 · · · dpN0
×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) ×
i
2
p10
1 1 i '(p10 − p10 ) · · · '(pN0 − pN0 ) : 0 2 pN − aN + i0 − a1 + i0
(95)
Integrating over the energies one easily obtains Eq. (47). 2.5.2. One-electron atom Formal expressions for the energy shift in the case of a one-electron atom (or in the case of an atom with one electron over closed shells) can be derived by various methods. In particular, the Dyson–Schwinger equation can be employed for such a derivation. Therefore, the one-electron system is not the best example to demonstrate the advantages of the method under consideration. However, we start with a detailed description of this simple case since it may serve as the simplest introduction to the technique. Let us consider ;rst a diagram describing the interaction of a one-electron atom with an external potential V (x) to ;rst order in V (x) (Fig. 11). According to Eq. (56) we have 2 i n1 (x ) P n1 (z) (1) d x d z d x a† (x ) (E)'(E − E ) = Tgaa i 2 n E − n1 (1 − i0) 1
×
2 0 i n2 (z) P n2 (x) 0 V (z)'(E − E) i 2 n E − n2 (1 − i0)
a (x)
2
= a| n1
=
|n1 n1 | |n2 n2 | V |a'(E − E) E − n1 (1 − i0) E − (1 − i 0) n 2 n 2
1 a|V |a'(E − E) : (E − a )2
Substituting (96) into (51), we obtain 1 (1) (1) TEa = d E(E − a )Tgaa (E) 2 i 4 1 a|V |a = dE = a|V |a : 2 i 4 E − a
(96)
(97)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
141
Fig. 11. The interaction with an external potential V (x). Fig. 12. The ;rst order self-energy diagram.
To ;rst order in , the QED corrections are de;ned by the self-energy (SE) and vacuum polarization (VP) diagrams which are shown in Figs. 12 and 13, respectively. Consider ;rst the SE diagram. We ;nd (1) Tgaa (E)'(E
2 −E )= i
×
d x d y d z d x
i
2
n1
† a (x )
∞ P (y) i ∞ n1 0 i dp d! 2 −∞ E − n1 (1 − i0) 2 −∞ n1 (x
)
×e(
P 2 n (y) n (z) '(E − p0 − !) D (!; y − z) 0 − (1 − i0) () i p n n
×e)
2 i n2 (z) P n2 (x) 0 '(p0 + ! − E) i 2 n E − n2 (1 − i0)
a (x)
2
1 i = e2 2 (E − a ) 2
∞
−∞
×D() (!; y − z) )
d!
dy dz
a (z)'(E
† ( a (y)
− E) ;
n
P
n (y) n (z)
E − ! − n (1 − i0) (98)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 13. The ;rst-order vacuum-polarization diagram. Fig. 14. The mass-counterterm diagram.
where ( ≡ 0 ( = (1; ). Denoting a|>(E)|b =
i
2
∞
−∞
d!
an|e2 ( ) D() (!)|nb n
E − ! − n (1 − i0)
;
(99)
we get (1) (E) = Tgaa
a|>(E)|a : (E − a )2
(100)
Substituting (100) into (51), we obtain 1 (1) d E(E − a )Tgaa (E) TEa(1) = 2 i 4 1 = 2 i
4
dE
a|>(E)|a = a|>(a )|a : E − a
(101)
Here we have taken into account the fact that, for a non-zero photon mass , Tg(E) has isolated poles at E = a in every order of perturbation theory. In the ;nal expression one can put → 0. Expression (101) suIers from an ultraviolet divergence and has to be considered together with a counterterm diagram (Fig. 14). Taking into account the counterterm results in a replacement a|>(a )|a → a|>R (a )|a = a|(>(a ) − 0 m)|a :
(102)
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143
The corresponding calculation for the VP diagram (Fig. 13) yields ∞ 2 i i 1 (1) d x d y P a (x) (E)'(E − E ) = − d ! Tgaa i 2 E − a 2 −∞ 2 2 '(E + ! − E)D() (! ; x − y)e '(! ) i i
∞ n (y) P (y) 1 i i n × d ! Tr ) 2 −∞ ! − n (1 − i0) 2 E − a n ×e(
a (x)
:
(103)
Introducing the VP potential by e2 UVP (x) = 2 i
(
d y D() (0; x − y)
∞
−∞
d ! Tr
n
† n (y) n (y)
! − n (1 − i0)
)
;
(104)
we ;nd (1) (E) = Tgaa
a|UVP |a (E − a )2
(105)
and, therefore, TEa(1) = a|UVP |a :
(106)
In reality, due to a spherical symmetry of the Coulomb potential of the nucleus, only zeroth components of the matrices contribute to UVP , ∞ 1 dy d ! Tr [GC (!; y; y)] ; (107) UVP (x) = 2 i |x − y| −∞ where GC (!; x; y) =
n
† n (x) n (y)
! − n (1 − i0)
(108)
is the Dirac–Coulomb Green’s function. Expression (107) is ultraviolet divergent. The charge renormalization makes it ;nite. Let us now consider the combined V -SE corrections described by the Feynman diagrams presented in Fig. 15. For the diagrams “a” and “b” one easily ;nds (2; a+b) (E) = Tgaa
1 1 a|V |n n|>(E)|a 2 (E − a ) n E − n
+
1 1 a|>(E)|n n|V |a : 2 (E − a ) n E − n
(109)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 15. V -self-energy diagrams.
This contribution is conveniently divided into two parts: irreducible (n = a ) and reducible (n = a ). For the irreducible part one obtains 1 (2; a+b; irr) d E(E − a )Tgaa (E) 2 i 4
(n=a) 1 a|V |nn|>(E)|a a|>(E)|nn|V |a 1 dE + = 2 i 4 E − a n E − n E − n (n=a)
=
n
a|V |nn|>(a )|a a|>(a )|nn|V |a + a − n a − n
:
(110)
Here we have taken into account that, due to the spherical symmetry of the Coulomb potential, matrix elements a|>(a )|b are equal to zero if a = b and a = b. The reducible part is 1 (2; a+b; red) d E(E − a )Tgaa (E) 2 i 4 1 1 dE = [a|V |aa|>(E)|a + a|>(E)|aa|V |a] 2 i 4 (E − a )2 = 2a|V |aa|> (a )|a ;
(111)
where > (a ) ≡ (d >(E)= d E)E=a . The reducible contribution should be considered together with the related contribution from the second term in Eq. (52). Taking into account that 1 (1; V ) d E(E − a )Tgaa (E) = a|V |a (112) 2 i 4
V.M. Shabaev / Physics Reports 356 (2002) 119–228
and 1 2 i
4
(1; SE) d ETgaa (E)
1 = 2 i
4
dE
a|>(E)|a = a|> (a )|a ; (E − a )2
145
(113)
we obtain 1 1 (1; V ) (1; SE) − d E(E − a )Tgaa (E) d ETgaa (E) = −a|V |aa|> (a )|a : 2 i 4 2 i 4 (114) For the total contribution of the diagrams “a” and “b” we ;nd (n=a) a|V |nn|>(a )|a a|>(a )|nn|V |a (2; a+b) = + TEa a − n a − n n + a|V |aa|> (a )|a :
(115)
For the vertex contribution (the diagram “c”) we obtain ∞ † 1 n1 (y) n1 (x) (2; c) 2 i † ( Tgaa (E) = d e d ! x d y d z (y) V (x) a (E − a )2 2 −∞ E − ! − n1 (1 − i0) n 1
× n2
† n2 (x) n2 (z)
E − ! − n2 (1 − i0)
D() (!; y − z) )
a (z)
:
(116)
This diagram is irreducible. A simple evaluation yields ∞ (2; c) 2 i TEa = e d ! d x d y d z a† (y) ( GC (a − !; y; x)V (x) 2 −∞ ×GC (a − !; x; z)D() (!; y − z) )
a (z)
:
(117)
The related mass-counterterm diagrams (Fig. 16) are accounted for by the replacement > → >R => − 0 m in Eq. (115). This replacement makes the irreducible contribution in Eq. (115) to be ;nite. As to the reducible contribution, its ultraviolet and infrared divergences are cancelled by the corresponding divergences of the vertex contribution given by Eq. (117). 2.5.3. Atom with one electron over closed shells The consideration given above can easily be adopted to the case of an atom with one electron over closed shells by regarding the closed shells as belonging to a rede;ned vacuum. The rede;nition of the vacuum results in replacing i0 by −i0 in the electron propagator denominators corresponding to the closed shells. In other words, it means replacing the standard Feynman contour of integration over the electron energy C by a new contour C (Fig. 17). In this formalism the one-electron radiative corrections are incorporated together with the interelectronic-interaction corrections and the total energy of the closed shells is considered as the origin of reference. The diIerence of the integrals along C and C is an integral along the contour Cint . It describes the interaction of the valence electron with the closed-shell electrons. Therefore, to ;nd the interelectronic-interaction corrections we have to replace the contour C in the expressions
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 16. V -mass-counterterm diagrams.
Fig. 17. C is the contour of the integration over the electron energy ! in the formalism with the standard vacuum. C is the integration contour for the vacuum with the (1s)2 shell included. The integral along the contour Cint = C − C describes the interaction of the valence electron with the (1s)2 -shell electrons.
for the one-electron radiative corrections by the contour Cint . For example, in the case of one electron over the (1s)2 shell in a lithium-like ion, the ;rst order interelectronic-interaction corrections are obtained from the formulas for the SE and VP corrections derived above by the replacement n
(c =1s ) |nn| 2 → − '(! − 1s ) |cc| : ! − n (1 − i0) i c
(118)
As a result of this replacement, one obtains for the interelectronic-interaction correction TEa(1; int)
(c =1s )
=
[ac|I (0)|ac − ac|I (a − 1s )|ca] ;
(119)
c
where I (!) is de;ned by Eq. (92). In Ref. [38] this formalism was employed to derive formal expressions for the interelectronic-interaction corrections to the hyper;ne splitting in lithium-like ions (see Section 4.3.2).
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147
Fig. 18. The ;rst order self-energy diagrams for a two-electron atom.
Fig. 19. The ;rst order vacuum-polarization diagrams for a two-electron atom.
Fig. 20. One-photon exchange diagram.
2.5.4. Two-electron atom Let us consider now the energy shift of a single level (n) in a two-electron atom. To ;rst order in , in addition to the one-electron SE and VP contributions (Figs. 18 and 19) we have to consider the one-photon exchange diagram (Fig. 20). The derivation of the energy shift due to the SE and VP diagrams is easily reduced to the case of a one-electron atom by a simple integration over the energy variable of a disconnected electron propagator. Therefore, below we discuss only the one-photon exchange diagram. For simplicity, we assume that the unperturbed wave function of the state under consideration is the one-determinant function 1 un (x1 ; x2 ) = √ (−1)P Pa (x1 ) Pb (x2 ) : (120) 2 P The transition to the general case of a many-determinant function (38) causes no problem and can be done in the ;nal expression for the energy shift.
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According to Eq. (56), for the one-photon exchange diagram we have 2 ∞ 1 1 i (1) Tgnn = dp10 dp10 (−1)P 0 0 2 p1 − Pa + i0 E − p1 − Pb + i0 −∞ P ×
p10
1 1 PaPb|I (p10 − p10 )|ab : 0 − a + i0 E − p1 − b + i0
(121)
Formula (51) gives 2 ∞ 1 i 1 (1) d E TE dp10 dp10 (−1)P 0 TEn = 2 i 4 2 p1 − Pa + i0 −∞ P ×
E−
p10
1 1 1 0 0 − Pb + i0 p1 − a + i0 E − p1 − b + i0
×PaPb|I (p10 − p10 )|ab ;
(122)
where, as in (51), TE ≡ E − En(0) . Transforming 1 1 1 1 1 ; = + p10 − Pa + i0 E − p10 − Pb + i0 TE p10 − Pa + i0 E − p10 − Pb + i0 1 1 1 = 0 0 TE p1 − a + i0 E − p1 − b + i0 we obtain 1 TEn(1) = 2 i ×
1 dE TE 4
i
2
2
∞
−∞
1 1 + 0 0 p1 − a + i0 E − p1 − b + i0
dp10 dp10
(123)
;
(124)
(−1)P
P
1 1 + p10 − Pa + i0 E − p10 − Pb + i0 1 1 0 0 PaPb|I (p1 − p1 )|ab : × + p10 − a + i0 E − p10 − b + i0
(125)
The expression in the curly braces of (125) is a regular function of E inside the contour 4, if the photon mass is chosen as indicated above. A direct way to check this fact consists in integrating over p10 and p10 by using the apparent expression for the photon propagator given by Eq. (53). It can also be understood by observing that the integrand in this expression is the sum of terms which contain singularities from the electron propagators in p10 (p10 ) only above or only below the real axis (for real E). Therefore, in each term we can vary E in the complex E-plane within the contour 4, keeping the same order of bypassing the singularities in the p10 (p10 ) integration by moving slightly the contour of the p10 (p10 ) integration into the complex plane. (In contrast to that, the contour of the p10 (p10 ) integration in Eq. (122) is sandwiched
V.M. Shabaev / Physics Reports 356 (2002) 119–228
149
between two poles and, therefore, cannot be moved to the complex plane.) The branch points of the photon propagators are moved outside the contour 4 due to the non-zero photon mass. However, instead of investigating the analytical properties for every speci;c diagram, it is more convenient to use the general analytical properties of the Green’s function gnn (E) in the complex E-plane in every order of perturbation theory (see the related discussion above and Appendix B). (1) According to these properties, the function Tgnn (E) cannot have poles at the point E = En(0) of order higher than 2 and, therefore, the expression in the curly braces of (125) is a regular function of E at this point. So, we have to calculate the ;rst order residue at the point E = En(0) . We also stress that we do not need any apparent form for the analytical continuation of the expression in the curly braces to the complex E-plane since we calculate it only for real E, at the point E = En(0) , where the present expression is valid. Calculating the E residue we obtain 2 ∞ i 1 1 (1) 0 0 P dp1 dp1 (−1) + TEn = 2 p10 − Pa + i0 −(p10 − Pa ) + i0 −∞ P 1 1 PaPb|I (p10 − p10 )|ab : (126) + × 0 0 p1 − a + i0 −(p1 − a ) + i0 Taking into account the identity i 1 1 = '(x) ; + 2 x + i0 −x + i0 we ;nd TEn(1) =
(−1)P PaPb|I (Pa − a )|ab :
(127)
(128)
P
2.5.5. General rules for practical calculations Let us consider now some general remarks to the derivation given above. In order to perform ;rst the integration over E we have separated the singularity in TE by employing identities (123) and (124). Another way could consist in transforming the left-hand sides of Eqs. (123) and (124) by the identity p0
1 1 1 2 '(p0 − a ) = 0 − a + i0 E − p − b + i0 i TE 1 1 + 0 ; 0 p − a − i0 E − p − b + i0
(129)
where we have used Eq. (127). Using this identity allows one to separate contributions singular in 1=TE from non-singular ones. The singular contributions result only from the ;rst term on the right-hand side of Eq. (129). It can easily be understood by observing that the second term has both singularities in p0 above the real axis (for real E). It follows that the contour of the p0 integration in the expression for the energy shift can be moved slightly into the complex E-plane keeping the same order of bypassing the singularities. It means that we can vary E in the complex E-plane within the contour 4 and, therefore, the integrand is a regular function of
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 21. Two-photon exchange diagrams.
E within this contour. Identities like (129) are very useful in calculations of three- and more electron atoms (see Section 4.2.3). We also want to note that in all cases the order of the singularity in 1=TE is quite evident from the type of the diagram under consideration. If the diagram is irreducible, the factors 1=TE may come only from the initial and ;nal propagators. In this case the second term on the right-hand side of Eq. (129) does not contribute to the energy shift and, therefore, the derivation of the formal expression for the energy shift becomes trivial. For reducible diagrams the factors 1=TE arise also from internal electron propagators. We can formulate the following simple rule for deriving the energy shift from a certain diagram. Using identities like (123), (124) or (129), we separate all singularities in 1=TE and then integrate over E assuming that the rest is a regular function of E within the contour 4. As is discussed above, the order of the singularity is quite evident for every speci;c diagram and it is a simple task to separate the factor 1=TE to the right power. However, even if we separate this factor to a power which is larger or smaller than the real order of the singularity, it is impossible to miss the correct result. In the ;rst case (the power is larger than the real order of the singularity) the result of the calculation remains the same as in the case when we separate the factor 1=TE to the right power. In the second case (the power is smaller than the real order of the singularity) we obtain an in;nite result (∼ 1=0). It means that we should increase the power of the separated singularity and repeat the calculation until we get a ;nite result. 2.5.6. Two-photon exchange diagrams for the ground state of a helium-like atom The two-photon exchange diagrams are presented in Fig. 21. Here we derive the energy shift from these diagrams for the case of the ground state of a helium-like atom. The case of a single excited state of a two-electron atom is considered in detail in [36]. The wave function of the ground state is given by 1 u1 (x1 ; x2 ) = √ (−1)P Pa (x1 ) Pb (x2 ) : (130) 2 P The unperturbed energy is E1(0) = a + b , where a = b .
V.M. Shabaev / Physics Reports 356 (2002) 119–228
151
Consider ;rst the two-photon ladder diagram (Fig. 21a). For the ;rst term in (52) we have 3 ∞ i 1 (2) P TElad = d E TE (−1) dp10 dp10 d ! 2 i 4 2 −∞ P × PaPb|I (p10 − !)|n1 n2 n1 n2 |I (! − p10 )|ab n1 n2
× ×
p10
1 1 1 0 − Pa + i0 E − p1 − Pb + i0 ! − n1 (1 − i0)
1 1 1 : 0 0 E − ! − n2 (1 − i0) p1 − a + i0 E − p1 − b + i0
(131)
Let us divide this contribution into irreducible (n1 +n2 = a +b ) and reducible (n1 +n2 =a +b ) parts (2) (2; irred) (2; red) TElad = TElad + TElad :
(132)
Using identities (123) and (124) we obtain for the irreducible part 3 ∞ 1 1 1 i (2; irred) P d E TE Tg11 (E) = dE (−1) dp10 dp10 d ! 2 i 4 2 i 4 TE 2 −∞ P n1 +n2 =a +b
×
n1 ; n 2
PaPb|I (p10 − !)|n1 n2 n1 n2 |I (! − p10 )|ab
1 1 + × 0 0 p1 − Pa + i0 E − p1 − Pb + i0
1 1 ! − n1 (1 − i0) E − ! − n2 (1 − i0) 1 1 × : + p10 − a + i0 E − p10 − b + i0
×
(133)
The expression in the curly braces of (133) is a regular function of E inside the contour 4 if the photon mass is chosen as indicated above. (If it were not so, we would get an in;nite result; see the related discussion in the previous subsection.) Calculating the E residue we ;nd (2; irred) TElad
=
P
P
(−1)
i
2
∞
−∞
d!
×n1 n2 |I (! − a )|ab
n1 +n2 =a +b
PaPb|I (Pa − !)|n1 n2
n1 ; n 2
1 1 : (0) ! − n1 (1 − i0) E1 − ! − n2 (1 − i0)
(134)
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This derivation shows that the energy shift from an irreducible diagram is obtained by evaluation of the “usual S-matrix” element. For the numerical evaluation of (134) it is convenient to rotate the contour of the integration in the complex ! plane [71]. For the reducible contribution we have 3 ∞ 1 1 1 i (2; red) P d E TETg11 (E) = dE (−1) dp10 dp10 d ! 2 2 i 4 2 i 4 (TE) 2 −∞ P n1 +n2 =a +b
×
n1 ; n 2
PaPb|I (p10 − !)|n1 n2 n1 n2 |I (! − p10 )|ab
1 1 + × p10 − Pa + i0 E − p10 − Pb + i0 1 1 × + ! − n1 + i0 E − ! − n2 + i0 1 1 : + × p10 − a + i0 E − p10 − b + i0
(135)
The expression in the curly braces of (135) is a regular function within the contour 4. Calculating the E residue and taking into account that a = b , we get 1 2 i
4
dE
(2; red) TETg11 (E) = −
i
2
P
P
(−1)
n1 +n2 =2a ∞ n1 ; n 2
−∞
dp10 PaPb|I (p10 − a )|n1 n2
1 + ×n1 n2 |I (0)|ab (a − p10 + i0)2 ×n1 n2 |I (a − p10 )|ab
+
∞
−∞
∞
−∞
dp10 PaPb|I (0)|n1 n2
1 (a − p10 + i0)2
d ! PaPb|I (a − !)|n1 n2
× n1 n2 |I (! − a )|ab
1 (a − ! + i0)2
:
(136)
This contribution should be considered together with the second term in Eq. (52). As was obtained above, the ;rst factor in this term is 1 (1) d E TETg11 (E) = (−1)P PaPb|I (0)|ab : (137) 2 i 4 P
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153
A simple calculation of the second factor yields 1 2 i
4
dE
2 ∞ 1 i dE dp10 dp10 2 (TE) 2 4 −∞ 1 1 P × (−1) + p10 − a + i0 E − p10 − a + i0 P 1 1 PaPb|I (p10 − p10 )|ab + × p10 − a + i0 E − p10 − a + i0 ∞ 1 i P =− (−1) dp10 0 PaPb|I (p10 − a )|ab 2 2 P (p − − i 0) a −∞ 1 ∞ 1 0 0 + dp1 0 PaPb|I (p1 − a )|ab : (138) (p1 − a − i0)2 −∞
(1) Tg11 (E) =
1 2 i
For the total reducible contribution we obtain 1 (2; red) (2; red) TElad = d E TE Tg11 (E) 2 i 4 1 1 (1) (1) − d E TETg11 (E) d E Tg11 (E) 2 i 4 2 i 4 n1 +n2 =2a i P =− (−1) n1 ; n 2
P
2
×n1 n2 |I (! − a )|ab
∞
−∞
d ! PaPb|I (! − a )|n1 n2
1 : (! − a − i0)2
(139)
A similar calculation of the two-photon crossed-ladder diagram (Fig. 21b) gives i ∞ (2) P TEcross = (−1) d ! Pan2 |I (! − a )|n1 bn1 Pb|I (! − a )|an2 2 −∞ n ;n P 1
×
2
1 1 : ! − n1 (1 − i0) ! − n2 (1 − i0)
(140)
(2; red) contains an infrared divergent term which is cancelled by a related The contribution TElad (2) . In the individual contributions the infrared term (n1 = n2 = a ) from the contribution TEcross singularities are regularized by a non-zero photon mass . If the ladder and crossed-ladder contributions are merged by the common !-integration, the integral is convergent and we can put → 0 before the integration over ! (see [36] for details). However, to show how the calculation
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for a non-zero photon mass can be performed, let us calculate the reducible contribution for a ;nite . We have to calculate the integral ∞ i exp[i !2 − 2 + i0(r12 + r34 )] 1 I1 = d! : (141) 2 −∞ r12 r34 (! − i0)2 Using the identity exp[i
!2
−
2
2 + i0 r] = −
we obtain 2 i 1 I1 = − 2 r12 r34
∞
−∞
0
d!
∞
∞
0
sin(kr) ; − k 2 − 2 + i0)
(142)
sin(k(r12 + r34 )) 1 : 2 2 2 (! − k − + i0) (! − i0)2
(143)
dk k
dk k
(!2
Decomposing the denominator !2 − k 2 − 2 + i0 = (! −
k 2 + 2 + i0)(! +
k 2 + 2 − i0)
(144)
and integrating over !, we ;nd ∞ 1 1 sin(k(r12 + r34 )) dk k : I1 = − r12 r34 0 (k 2 + 2 )3=2
(145)
According to [72], the last integral is I1 = −
1 r12 + r34 K0 [ (r12 + r34 )] ; r12 r34
(146)
where ∞ ∞ (z=2)2k z 2k + (k + 1) : K0 (z) = −log(z=2) (k!)2 22k (k!)2 k=0
(147)
k=0
Considering → 0 we ;nd (2; red) =− TElad
2 (−1)P P
n1 =n2 =a
×
n1 ; n 2
n1 (x3 )
d x1 · · · d x4 P Pa (x3 ) P Pb (x4 )(3 )4
P (x1 ) n1
$ P n2 (x4 ) n2 (x2 )1 2
×[log(r12 + r34 ) + log − log 2 − (1)]g() g$
1 1 + r12 r34
a (x1 ) b (x2 )
:
(148)
The corresponding contribution (n1 = n2 = a ) from the crossed-ladder diagram is calculated in the same way. The sum of the reducible contribution from the ladder diagram and the related
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155
contribution from the crossed-ladder diagram is [29] n1 =n2 =a 2 (2; infr) P d x1 · · · d x4 P Pa (x3 ) P Pb (x4 )(3 )4 =− (−1) TE P n ;n 1
2
P (x1 ) n2 (x4 ) P (x2 )1 $2 n1 n2 1 1 log(r12 + r34 ) × g() g$ + r12 r34
1 1 log(r14 + r23 ) −g($ g) + r14 r23 ×
n1 (x3 )
a (x1 ) b (x2 )
:
(149)
The terms containing the factor log −log 2 − (1) have cancelled each other. For the numerical evaluation of this contribution, one can use the following representation [36]: 1 ∞ dy TE (2; infr) = ba|S (y)|ab[2ab|S(y)|ab − aa|S(y)|aa − bb|S(y)|bb] 0 y 1 ∞ dy + ba|S(y)|ab[2ab|S (y)|ab − aa|S (y)|aa − bb|S (y)|bb] 0 y 2 ∞ dy − ba|S(y)|abba|S (y)|ab ; (150) 0 y where S(y) =
(1 − 1 · 2 ) −yr12 e ; r12
(151)
S (y) = − (1 − 1 · 2 )e−yr12 :
(152)
Other representations of this term can be found in Refs. [71,73]. 2.5.7. Self-energy and vacuum-polarization screening diagrams The self-energy and vacuum-polarization screening diagrams are presented in Figs. 22 and 23, respectively. The derivation of the calculation formulas for the energy shift of a single level due to the SE screening diagrams using the TTGF method is described in detail in Ref. [43]. If the unperturbed wave function is given by Eq. (120), the contribution of the SE screening diagrams is written as TE = TE irred + TE red + TE ver ; n =a (−1)P PaPb|I (B)|nb TE irred = n
P n =b
+
n
PaPb|I (B)|an
(153) 1 n|>(a )|a a − n
1 n|>(b )|b b − n
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Fig. 22. The self-energy screening diagrams. n =Pa
+
n n =Pb
+
n
Pa|>(Pa )|n
1 nPb|I (B)|ab Pa − n
1 Pb|>(Pb )|n Pan|I (B)|ab ; Pb − n
(154)
TE red = ba|I (b − a )|ab [a|>(a )|a − b|>(b )|b] +
(−1)P PaPb|I (B)|ab[a|> (a )|a + b|> (b )|b] ;
(155)
P
TE
ver
=
P
P
(−1)
i n1 n2
2
∞
−∞
d!
n1 Pb|I (B)|n2 bPan2 |I (!)|n1 a [Pa − ! − n1 (1 − i0)][a − ! − n2 (1 − i0)]
Pan1 |I (B)|an2 Pbn2 |I (!)|n1 b + [Pb − ! − n1 (1 − i0)][b − ! − n2 (1 − i0)]
;
(156)
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157
Fig. 23. The vacuum-polarization screening diagrams.
where B = Pa − a . The corresponding contribution of the VP screening diagrams is [40,44] TE = TEairred + TEared + TEb ;
(157)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
TEairred
=
n =a
P
(−1)
n
P
+
PaPb|I (B)|nb
n =b
PaPb|I (B)|an
n
+
n =Pa
Pa|UVP |n
n
+
n =Pb n
1 n|UVP |a a − n
1 n|UVP |b b − n
1 nPb|I (B)|ab Pa − n
1 Pb|UVP |n Pan|I (B)|ab ; Pb − n
TEared = ba|I (b − a )|ab[a|UVP |a − b|UVP |b] ; TEb =
(−1)P PaPb|IVP (B)|ab ;
(158)
(159) (160)
P
where 2 IVP (; x; y) = 2 i
∞
1 exp(i|||x − z1 |) 2$ exp(i|||y − z2 |) |x − z1 | |y − z2 | −∞ : ×Tr G ! − ; z1 ; z2 $ G ! + ; z2 ; z1 2 2 d!
d z1
d z2
(161)
Expressions (153) – (160) are ultraviolet divergent. The renormalization of these expressions can be performed in the same way as for the ;rst-order SE and VP contributions (see Refs. [39,40,42,43] for details). 2.5.8. Quasi-degenerate states Let us now consider some applications of the method to the case of quasi-degenerate states of a helium-like ion. This case arises, for instance, if one is interested in the energies of the (1s2p1=2 )1 and (1s2p3=2 )1 states. These states are strongly mixed for low and middle Z and, therefore, must be treated as quasi-degenerate. It means that the oI-diagonal matrix elements of the energy operator H between these states have to be taken into account. The unperturbed wave functions are written as ui (x1 ; x2 ) = AN ji1 mi1 ji2 mi2 |JM (−1)P Pi1 (x1 ) Pi2 (x2 ) ; (162) mi1 mi2
P
√ where AN is the normalization factor equal to 1= 2 for non-equivalent electrons and to 1=2 for equivalent electrons, J is the total angular momentum, and M is its projection. However, in
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159
what follows, to compactify the formulas we will construct the matrix elements of H between the one-determinant wave functions 1 ui (x1 ; x2 ) = √ (−1)P Pi1 (x1 ) Pi2 (x2 ) : (163) 2 P The transition to the wave functions de;ned by Eq. (162) can easily be accomplished in the ;nal formulas. First we consider the contribution from the one-photon exchange diagram. To derive the formulas for Hik(1) we will assume that Ei(0) = Ek(0) . However, all the ;nal formulas remain to be valid also for the case Ei(0) = Ek(0) which was considered in detail above. According to the Feynman rules and the de;nition of g(E), the contribution of the one-photon exchange diagram to g(1) (E) is 2 ∞ 1 i 1 (1) dp1 dp10 (−1)P 0 gik (E) = 0 2 p1 − Pi1 + i0 E − p1 − Pi2 + i0 −∞ P 1 1 Pi1 Pi2 |I (p10 − p10 )|k1 k2 : (164) 0 − k1 + i0 E − p1 − k2 + i0 Using identities (123) and (124), we obtain i 2 ∞ 1 E (1) 0 0 dE d p d p (−1)P Kik = 1 1 2 i 4 (E − Ei(0) )(E − E (0) ) 2 −∞ P k 1 1 1 1 × + + p10 − Pi1 + i0 E − p10 − Pi2 + i0 p10 − k1 + i0 E − p10 − k2 + i0 ×
p10
× Pi1 Pi2 |I (p10 − p10 )|k1 k2
:
(165)
The expression in the curly braces of (165) is a regular function of E inside the contour 4, if the photon mass is chosen as indicated above. Calculating the E residues and taking into account identity (127), we obtain ∞ (0) 0 i (1) 0 P Ei Pi1 Pi2 |I (Pi1 − p1 )|k1 k2 Kik = dp1 (−1) 2 −∞ Ei(0) − Ek(0) P 1 1 × + p10 − k1 + i0 Ei(0) − p10 − k2 + i0 +
×
i
2
∞
−∞
p10
dp10
P
(−1)P
Ek(0) Pi1 Pi2 |I (p10 − k1 )|k1 k2 Ek(0) − Ei(0)
1 1 + (0) 0 − Pi1 + i0 Ek − p1 − Pi2 + i0
:
(166)
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In the same way we ;nd ∞ Pi1 Pi2 |I (Pi1 − p10 )|k1 k2 i dp10 (−1)P Pik(1) = 2 −∞ Ei(0) − E (0) P k
×
+
1 1 + (0) 0 p1 − k1 + i0 Ei − p10 − k2 + i0
i
2
×
∞
−∞
dp10
P
(−1)P
Pi1 Pi2 |I (p10 − k1 )|k1 k2
Ek(0) − Ei(0)
1 1 + p10 − Pi1 + i0 Ek(0) − p10 − Pi2 + i0
:
(167)
Symmetrizing Eqs. (166) and (167) with respect to both electrons we transform them to the form 1 (1) P Kik = (−1) [Pi1 Pi2 |I (B1 )|k1 k2 + Pi1 Pi2 |I (B2 )|k1 k2 ] 2 P ∞ (Ei(0) + Ek(0) ) i − d !Pi1 Pi2 |I (!)|k1 k2 2 2 −∞
1 1 × ; + (! + B1 − i0)(! − B2 − i0) (! + B2 − i0)(! − B1 − i0) ∞ i Pik(1) = − (−1)P d !Pi1 Pi2 |I (!)|k1 k2 2 −∞ P
1 1 × + (! + B1 − i0)(! − B2 − i0) (! + B2 − i0)(! − B1 − i0)
(168)
;
(169)
where B1 = Pi1 − k1 and B2 = Pi2 − k2 . Substituting (168), (169) into (86), we get [35,74] 1 Hik(1) = (−1)P [Pi1 Pi2 |I (B1 )|k1 k2 + Pi1 Pi2 |I (B2 )|k1 k2 ] : (170) 2 P Let us now consider the contribution to H from the combined V -interelectronic-interaction diagrams presented in Fig. 24. For simplicity, we will assume that V is a spherically symmetric potential. In the case under consideration, the simplest way to derive the formulas for Hik(2) consists in using the fact that these diagrams can be obtained as the ;rst order (in V ) correction to the one-photon exchange contribution derived above. So, the contribution from these diagrams can be obtained by the following replacements in Eq. (170): |k1 → |k1 + '|k1 ;
(171)
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161
Fig. 24. V -interelectronic-interaction diagrams.
|k2 → |k2 + '|k2 ;
(172)
|Pi1 → |Pi1 + '|Pi1 ;
(173)
|Pi2 → |Pi2 + '|Pi2 ;
(174)
I (a − b ) → I (a + 'a − b − 'b ) ;
(175)
where, to ;rst order in V , 'a = a|V |a ; '|a =
n =a n
|nn|V |a : a − n
(176) (177)
Here we have taken into account that, due to the spherical symmetry of V , n|V |a = 0 if n = a and |n = |a. Decomposing the modi;ed expression for the one-photon exchange diagram to the ;rst order in V , we ;nd that the total correction is the sum of the irreducible and reducible parts, Hik(2) = Hik(2; irred) + Hik(2; red) ; where Hik(2; irred) =
1 (−1)P ['Pi1 Pi2 |I (B1 ) + I (B2 )|k1 k2 + Pi1 'Pi2 |I (B1 ) + I (B2 )|k1 k2 2 P + Pi1 Pi2 |I (B1 ) + I (B2 )|'k1 k2 + Pi1 Pi2 |I (B1 ) + I (B2 )|k1 'k2 ]
and Hik(2; red) =
(178)
(179)
1 (−1)P {[Pi1 |V |Pi1 − k1 |V |k1 ]Pi1 Pi2 |I (B1 )|k1 k2 2 P + [Pi2 |V |Pi2 − k2 |V |k2 ]Pi1 Pi2 |I (B2 )|k1 k2 } :
(180)
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Eqs. (179) and (180) provide the matrix elements between the one-determinant wave functions de;ned by Eq. (163). To get the matrix elements between the wave functions de;ned by Eq. (162), we have to multiply these equations by the Clebsch–Gordan coeKcients and to sum over the projections of the one-electron angular momenta. The expression for Hik(2) can also be derived by direct application of the TTGF method. It can easily be performed in the same way as for the one-photon exchange diagram. We note that this derivation yields a formula for Hik(2) which is slightly diIerent from the expression given above. In particular, for the irreducible contribution one ;nds Hik(2; irred) =
1 (−1)P ['Pi1 Pi2 |I (B1 ) + I (B2 )|k1 k2 2 P + Pi1 Pi2 |I (B1 ) + I (B2 )|k1 k2 + Pi1 Pi2 |I (B1 ) + I (B2 )|'k1 k2 + Pi1 Pi2 |I (B1 ) + I (B2 )|k1 'k2 ] + BHik(2; irred) ;
where BHik(2; irred)
∞ i 1 = (Ei(0) − Ek(0) ) (−1)P d! 2 2 −∞ P
+ − −
+
Pi1 Pi2 |I (! − k1 )|'k1 k2
(! − Pi1 + i0)(Ek(0) − ! − Pi2 − i0) Pi1 Pi2 |I (! − k1 )|k1 'k2
(! − Pi1 − i0)(Ek(0) − ! − Pi2 + i0) n =k1
Pi1 Pi2 |I (! − Pi1 )|nk2 n|V |k1
(! − k1 − i0)(Ei(0) − ! − k2 + i0)(! − n (1 − i0))
n =k2
n =Pi1
n =Pi2 n
Pi1 Pi2 |I (! − Pi1 )|k1 nn|V |k2
(! − k1 + i0)(Ei(0) − ! − k2 − i0)(Ei(0) − ! − n (1 − i0))
n
−
(! − k1 + i0)(Ei(0) − ! − k2 − i0)
(! − k1 − i0)(Ei(0) − ! − k2 + i0)
n
−
'Pi1 Pi2 |I (! − Pi1 )|k1 k2
Pi1 Pi2 |I (! − Pi1 )|k1 k2
n
+
(181)
Pi1 |V |nnPi2 |I (! − k1 )|k1 k2
(! − Pi1 − i0)(Ek(0) − ! − Pi2 + i0)(! − n (1 − i0)) Pi2 |V |nPi1 n|I (! − k1 )|k1 k2
(! − Pi1 + i0)(Ek(0) − ! − Pi2 − i0)(Ek(0) − ! − n (1 − i0))
:
(182)
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163
The term THik(2; irred) approaches zero if Ei(0) → Ek(0) . Expressions (179) and (181) diIer by the term THik(2; irred) which can be represented as THik(2; irred) = (Ei(0) − Ek(0) )Oik(2) :
(183)
Here O(2) is an operator of second order in the perturbation parameter which we denote by 0 (for simplicity, we assume here that V and the interelectronic-interaction operator I (!) are characterized by the same perturbation parameter). This fact can be understood by observing that the integrand in (182) is the sum of terms which contain singularities from the external electron propagators only above or below of the real axis and, therefore, integrating over ! cannot result in the appearance of contributions ∼1=(Ei(0) − Ek(0) ) that could compensate the factor (Ei(0) − Ek(0) ). In particular, it means that Oik(2) remains ;nite when Ei(0) → Ek(0) . It can be shown that the term THik(2; irred) contributes only to third and higher orders in 0 and, therefore, can be omitted if we restrict our calculations to second order in 0 . Let us prove this fact for the case of two quasi-degenerate levels. In this case the energy levels are determined from the equation (E − H11 )(E − H22 ) − H12 H21 = 0
(184)
which yields E1; 2 =
H11 + H22 1 ± (H11 − H22 )2 + 4H12 H21 : 2 2
(185)
If E1 − E2 ∼ 0 , the proof of the statement is evident. If E1 − E2 0 , the contribution of the second-order oI-diagonal matrix elements is given by (2) (1) (1) (2) TE1; 2 ≈ ± (H12 H21 + H12 H21 )=(E1(0) − E2(0) ) :
(186)
From this equation we ;nd that the terms ∼ (Ei(0) − Ek(0) )Oik(2) in Hik(2) contribute only to third and higher orders in 0 . Eqs. (178) – (180) will give a part of the VP screening contribution, if we replace V by UVP . The total contribution of the VP screening diagrams for quasi-degenerate states is derived in Ref. [44]. Corresponding formulas for the SE screening diagrams are derived in Ref. [75]. 2.6. Nuclear recoil corrections So far we considered the nucleus as a source of the external Coulomb ;eld VC . This consideration corresponds to the approximation of an in;nite nuclear mass. However, high precision calculations of the energy levels in high-Z few-electron atoms must include also the nuclear recoil corrections to ;rst order in m=M (M is the nuclear mass) and to zeroth order in (but to all orders in Z). As was shown in [41], these corrections can be included in calculations of the energy levels by adding an additional term to the standard Hamiltonian of the electron–positron ;eld interacting with the quantized electromagnetic ;eld and with the Coulomb ;eld of the
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
nucleus VC . In the Coulomb gauge, this term is given by 1 † HM = d x (x)(−i∇x ) (x) d y † (y)(−i∇y ) (y) 2M e2 Z 2 2 eZ d x † (x)(−i∇x ) (x)A(0) + − (187) A (0) : M 2M HM taken in the interaction representation must be added to the interaction Hamiltonian. It gives the following additional lines and vertices to the Feynman rules (we assume that the Coulomb gauge and the Furry picture are used). 1. Coulomb contribution: An additional line (“Coulomb-recoil” line) appears to be
i 'kl
∞
d! : 2 M −∞ This line joins two vertices each of which corresponds to
0
−2 i '(!1 − !2 − !3 )
d x pk ;
where p = −i∇x and k = 1; 2; 3. 2. One-transverse-photon contribution: An additional vertex on an electron line appears to be
eZ d x pk : M The transverse photon line attached to this vertex (at the point x) is −2 i0 '(!1 − !2 − !3 )
i
2
∞
−∞
d ! Dkl (!; y) :
At the point y this line is to be attached to a usual vertex, where we have −2 ie0 l 2 ' × (!1 − !2 − !3 ) d y. l (l = 1; 2; 3) are the usual Dirac matrices. 3. Two-transverse-photon contribution: An additional line (“two-transverse-photon-recoil” line) appears to be i e2 Z 2
∞
d ! Dil (!; x)Dlk (!; y) : 2 M −∞ This line joins usual vertices (see the previous item).
V.M. Shabaev / Physics Reports 356 (2002) 119–228
165
Fig. 25. Coulomb nuclear recoil diagram. Fig. 26. One-transverse-photon nuclear recoil diagrams.
Let us apply this formalism to the case of a single level a in a one-electron atom. To ;nd the Coulomb nuclear recoil correction, we have to calculate the contribution of the diagram shown in Fig. 25. According to the Feynman rules given above we obtain ∞ a|pi |nn|pi |a 1 1 i (1) Tgaa (E) = d ! : (188) ! − n (1 − i0) (E − Ea(0) )2 M 2 −∞ n Formula (51) gives 1 i TEC = M 2
∞
−∞
d!
a|pi |nn|pi |a n
! − n (1 − i0)
:
(189)
The one-transverse-photon nuclear recoil correction corresponds to the diagrams shown in Fig. 26. A similar calculation yields ∞ a|pi |nn| k Dik (a − !)|a 4 Z i d! TEtr(1) = M 2 −∞ ! − n (1 − i0) n a| k Dik (a − !)|nn|pi |a + ! − n (1 − i0)
:
(190)
The two-transverse-photon nuclear recoil correction is de;ned by the diagram shown in Fig. 27. We ;nd ∞ a| i Dil (a − !)|nn| k Dlk (a − !)|a (4 Z)2 i TEtr(2) = d! : (191) M 2 −∞ ! − n (1 − i0) n The sum of all the contributions is ∞ 1 i d !a|[pi + 4 Z l Dli (!)]GC (! + a )[pi + 4 Z m Dmi (!)]|a ; TE = M 2 −∞
(192)
where GC (!) is the Dirac–Coulomb Green’s function de;ned above. For practical calculations it is convenient to represent expression (192) by the sum of a lower-order term TEL and a higher-order term TEH : TE = TEL + TEH ;
(193)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 27. Two-transverse-photon nuclear recoil diagram. Fig. 28. Two-electron Coulomb nuclear recoil diagram.
Fig. 29. Two-electron one-transverse-photon nuclear recoil diagrams. Fig. 30. Two-electron two-transverse-photon nuclear recoil diagram.
1 a|[pi2 − (Di (0)pi + pi Di (0))]|a ; 2M ∞ [pi ; VC ] i [pi ; VC ] |a ; d !a| Di (!) − GC (! + a ) Di (!) + TEH = 2 M −∞ ! + i0 ! + i0
TEL =
(194) (195)
where Di (!) = −4 Z l Dli (!). The term TEL contains all the recoil corrections within the ( Z)4 m2 =M approximation, while the term TEH contains the contribution of order ( Z)5 m2 =M and all contributions of higher orders in Z which are not included in TEL . Formulas (193) – (195) were ;rst derived by a quasi-potential method in Ref. [50] and subsequently rederived by other methods in Refs. [76,77]. Representation (192) was found in Ref. [76]. Consider now a two-electron atom. For simplicity, as usual, we assume that the unperturbed wave function is a one-determinant function (120). The nuclear recoil correction is the sum of the one- and two-electron contributions. Using the Feynman rules and formula (51), one easily ;nds that the one-electron contribution is equal to the sum of expressions (192) for the a and b states. The two-electron contributions correspond to the diagrams shown in Figs. 28–30. A simple calculation of these diagrams yields 1 TE (int) = (−1)P Pa|[pi + 4 Z l Dli (Pa − a )]|a M P ×Pb|[pi + 4 Z m Dmi (Pb − b )]|b :
Formula (196) was ;rst derived by a quasi-potential method in Ref. [51].
(196)
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167
3. Transition probabilities and cross sections of scattering processes According to the basic principles of quantum ;eld theory [78], the number of the particles scattered into the interval d p1 · · · d pr for a unit time and in a unit volume is d Wp→p = (2 )3s+1 n1 · · · ns |Gpp |2 '(Ep − Ep ) d p1 · · · d pr ;
(197)
where p; p are the initial and ;nal states of the system, respectively; Gpp is the amplitude of the process de;ned by p |(S − I )|p = 2 i'(Ep − Ep )Gpp ;
(198)
S is the scattering operator, I is the identity operator, s is the number of initial particles, r is the number of ;nal particles; n1 ; : : : ; ns are the average numbers of particles per unit volume. We will consider the scattering of photons and electrons by an atom that is located at the origin of the coordinate frame. The diIerential cross section is de;ned by d) =
d Wp→p
j
;
(199)
where j is the current of initial particles (for photons j = nc; for electrons j = nv, where v is the velocity of the electrons in the nucleus frame). The total cross section can be found by integrating the diIerential cross section over all ;nal states. The cross section for the elastic scattering is (2 )4 (elast) d p '(Ep − Ep )|Gpp |2 : )tot (p) = (200) v The total (elastic plus inelastic) cross section can be found by employing the optical theorem 2(2 )3 Im G pp : (201) v In terms of the amplitude fpp which is de;ned so that d ) = |fpp |2 d 9, the optical theorem has a well-known form (see, e.g., [79]) )tot (p) =
)tot (p) =
4 Im fpp : |p|
(202)
The aim of this section is to derive formulas for the transition and scattering amplitudes for various processes in the framework of QED. 3.1. Photon emission by an atom Consider the process of the photon emission by an atom. According to the standard reduction technique (see, e.g., [28,56]), the atomic transition amplitude from states a to b accompanied by photon emission with momentum kf and polarization if is j$∗ f exp (ikf · y) Sf ;b;a = b|aout (kf ; jf )|a = −iZ3−1=2 d 4 y b|j$ (y)|a : (203) 2kf0 (2 )3
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Here j$ (y) is the electron–positron current operator in the Heisenberg representation, |a and |b are the vectors of the initial and ;nal states in the Heisenberg representation, Z3 is a renormalization constant, a · b ≡ a$ b$ , jf = (0; if ), kf = (kf0 ; kf ), and kf0 ≡ |kf |. Employing the equation (see Appendix A) j $ (y) = exp (iHy0 )j $ (0; y) exp (−iHy0 ) ; we obtain Sf ;b;a = −iZ3−1=2
(204)
d 4 y exp[i(Eb + kf0 − Ea )y0 ]A$∗ f (y)b|j$ (0; y)|a = −2 iZ3−1=2 '(Eb + kf0 − Ea ) d y A$∗ f (y)b|j$ (0; y)|a ;
(205)
j$f exp(ikf · x) =
(206)
where A$f (x)
2kf0 (2 )3
is the wave function of the emitted photon. Since |a and |b are bound states, Eq. (205) as well as the standard reduction technique [28,56] cannot be used for a direct evaluation of the amplitude. The desired calculation formula can be derived within the two-time Green’s function formalism [30 –32]. To formulate the method for a general case, we assume that in zeroth approximation the state a belongs to an sa -dimensional subspace of unperturbed degenerate states 9a and the state b belongs to an sb -dimensional subspace of unperturbed degenerate states 9b . We denote the projectors onto these subspaces by Pa(0) and Pb(0) , respectively. We denote the exact states originating from 9a by |na and the exact states originating from 9b by |nb . We also assume that on an intermediate stage of the calculations a non-zero photon mass is introduced. It is considered to be larger than the energy splitting of the initial and ;nal states under consideration and much smaller than the distance to other levels. We introduce Gf (E ; E; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E + k 0 − E)
1 1 1 = 2 i 2 N !
∞
−∞
0
dx dx
0
d 4 y exp(iE x0 − iEx0 ) exp(ik 0 y0 )
0 0 P 0 P 0 × A$∗ f (y)0|T (x ; x1 ) · · · (x ; xN )j$ (y) (x ; xN ) · · · (x ; x1 )|0 ;
(207)
where, as in the previous section, (x) is the electron–positron ;eld operator in the Heisenberg representation. Let us investigate the singularities of Gf in the vicinity of the points E ≈ Eb(0) and E ≈ Ea(0) . Using the transformation rules (x0 ; x) = exp(iHy0 ) (x0 − y0 ; x) exp(−iHy0 ); j(y0 ; y) = exp(iHy0 )j(0; y) exp(−iHy0 ) ;
(208)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
169
we obtain Gf (E ; E; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E + k 0 − E)
1 1 1 = 2 i 2 N !
∞
−∞
dt dt
d 4 y exp(iE t − iEt) exp[i(E + k 0 − E)y0 ]
P P ×A$∗ f (y)0|T (t ; x1 ) · · · (t ; xN )j$ (0; y) (t; xN ) · · · (t; x1 )|0
1 1 = '(E + k 0 − E) 2 i N!
∞
−∞
dt dt
d y exp(iE t − iEt)
P P × A$∗ f (y)0|T (t ; x1 ) · · · (t ; xN )j$ (0; y) (t; xN ) · · · (t; x1 )|0 :
(209)
Using again the time-shift transformation rules, we obtain Gf (E ; E; x1 ; : : : ; xN ; x1 ; : : : ; xN )
=
1 1 2 i N !
∞
−∞
dt dt
d y exp(iE t − iEt)
n1 ; n 2
A$∗ f (y)
×exp(−iEn1 t ) exp(iEn2 t).(t ).(−t)0|T (0; x1 ) · · · (0; xN )|n1 ×n1 |j$ (0; y)|n2 n2 | P (0; xN ) · · · P (0; x1 )|0 + · · · :
(210)
Here we have assumed E0 = 0, as in the previous section. Taking into account the identities ∞ i ; d t exp[i(E − En1 )t] = E − En1 + i0 0
0
−∞
d t exp[i(−E + En2 )t] =
i
E − En2 + i0
;
(211)
we ;nd Gf (E ; E; x1 ; : : : ; xN ; x1 ; : : : ; xN )
=
i 1
2 N ! n ; n 1
2
d y A$∗ f (y)
E
1 1 − En1 + i0 E − En2 + i0
×0|T (0; x1 ) · · · (0; xN )|n1 n1 |j$ (0; y)|n2 ×n2 | P (0; xN ) · · · P (0; x1 )|0 + · · · :
(212)
We are interested in the analytical properties of Gf as a function of the two complex variables E and E in the region E ≈ Eb(0) , E ≈ Ea(0) . These properties can be studied using the double
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spectral representation of this type of Green’s function which is given in Appendix E (a similar representation was derived in Ref. [64]). As it follows from the spectral representation, the terms which are omitted in Eq. (212) are regular functions of E or E if E ≈ Eb(0) and E ≈ Ea(0) . Eq. (212) and the spectral representation given in Appendix E show that, for a non-zero photon mass , the Green’s function Gf (E ; E) has isolated poles in the variables E and E at the points E = Enb and E = Ena , respectively. Let us now introduce a Green’s function gf ;b;a (E ; E) by gf ;b;a (E ; E) = Pb(0) Gf (E ; E)01 · · · 0N Pa(0) ;
(213)
where, as in (39), the integration over the electron coordinates is implicit. According to Eq. (212) (see also Appendix E) the Green’s function gf ;b;a (E ; E) can be written as
gf ;b;a (E ; E) =
sb sa i
2
na =1nb =1
1 1 ’n E − Enb E − Ena b
† d y A$∗ f (y)nb |j$ (0; y)|na ’na
+ terms that are regular functions of E or E if E ≈ Eb(0) and E ≈ Ea(0) ; (214) where the vectors ’k are de;ned by Eq. (65). Let the contours 4a and 4b surround the poles corresponding to the initial and ;nal levels, respectively, and keep outside other singularities of gf ;b;a (E ; E) including the cuts starting from the lower-lying bound states. Comparing Eq. (214) with Eq. (205) and taking into account the biorthogonality condition (71), we obtain the desired formula [30] Sf ;b;a = Z3−1=2 '(Eb + kf0 − Ea ) dE d E vb† gf ;b;a (E ; E)va ; (215) 4b
4a
where by a we imply one of the initial states and by b one of the ;nal states under consideration. The vectors vk are determined from Eqs. (76) – (77). In the case of a single initial state (a) and a single ;nal state (b), the vectors va and vb simply become normalization factors. So, for the initial state, 1 d E gaa (E)va = 1 (216) va∗ Pa va = va∗ 2 i 4a and, therefore,
−1 1 |va |2 = d E gaa (E) : 2 i 4a
(217)
Choosing
1 va = 2 i
−1=2
4a
d E gaa (E)
;
1 vb = 2 i
−1=2
4b
d E gbb (E)
;
(218)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
we obtain Sf ;b;a = Z3−1=2 '(Eb + kf0 − Ea )
1 × 2 i
4b
dE
−1=2
4b
d E gbb (E)
1 2 i
4a
171
d E gf ;b;a (E ; E)
−1=2
4a
d E gaa (E)
:
(219)
For practical calculations of the transition amplitude, it is convenient to express the Green’s function gf ;b;a (E ; E) in terms of the Fourier transform of the 2N -time Green’s function, ∞ 1 gf ;b;a (E ; E)'(E + k 0 − E) = dp0 · · · dpN0 dp10 · · · dpN0 N ! −∞ 1 ×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) (0)
×Pb Gf (p10 ; : : : ; pN0 ; k 0 ; p10 ; : : : ; pN0 )01 : : : 0N Pa(0) ;
(220)
where Gf ((p10 ; x1 ); : : : ; (pN0 ; xN ); k 0 ; (p10 ; x1 ); : : : ; (pN0 ; xN )) ∞ 2 1 0 0 0 0 d x : : : d xN d x1 : : : d xN d 4 y = i (2 )2N +1 −∞ 1 × exp(ip10 x10 + · · · + ipN0 xN0 − ip10 x10 − · · · − ipN0 xN0 + ik 0 y0 ) P P × A$∗ f (y)0|T (x1 ) · · · (xN )j$ (y) (xN ) · · · (x1 )|0 :
(221)
The Green’s function Gf is constructed by perturbation theory after the transition in (221) to the interaction representation and using the Wick theorem. The Feynman rules for Gf differ from those for G considered in the previous section only by the presence of an outgoing photon line that corresponds to the wave function of the emitted photon A$∗ f (x). The Feynman rule for a vertex in which a real photon is emitted remains the same as for a virtual photon vertex. The energy variable of the emitted photon (k 0 ) in the expression corresponding to a real photon vertex is considered as a free variable (k 0 = kf0 = |kf |) which, due to the energy conservation, can be expressed in terms of the initial (E) and ;nal (E ) atomic energy variables. 3.2. Transition probability in a one-electron atom To demonstrate the practical ability of the formalism, in this section we derive formulas for the transition probability in a one-electron atom to zeroth and ;rst orders in . An application of the method for two-electron atoms is considered in [80].
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 31. The photon emission by a one-electron atom in zeroth order approximation.
3.2.1. Zeroth order approximation To zeroth order the transition amplitude is described by the diagram shown in Fig. 31. Formula (219) gives (0) (0) 0 Sf ;b;a = '(Eb + kf − Ea ) dE d E gf ;b;a (E ; E) ; (222) 4b
4a
where the superscript indicates the order in . Here we have taken into account that 1 1 1 (0) d E gaa (E) = dE =1 2 i 4a 2 i 4a E − a
(223)
and a similar equation exists for the b state. According to the Feynman rules, we have i 0 d y S(E ; x ; y)(−2 ie$ )'(E + k 0 − E) ((E ; x ); k ; (E; x)) = G(0) f 2 ×A∗f; $ (y)
i
2
S(E; y; x) :
(224)
Substituting expression (224) into the de;nition of gf ;b;a (E ; E) (see Eq. (220)), we obtain (E ; E) = g(0) f ;b;a
i
2
E
1 1 b|e $ A∗f; $ |a : − b E − a
(225)
Eqs. (222) and (225) yield S(0) = −2 i'(Eb + kf0 − Ea )b|e $ A∗f; $ |a f ;b;a
(226)
or, in accordance with de;nition (198), $ ∗ ∗ G(0) f ;b;a = −b|e Af; $ |a = b|e · Af |a :
(227)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
173
The transition probability to zeroth order is (0)
(0)
d Wf ;b;a = 2 |Gf ;b;a |2 '(Eb + kf0 − Ea ) d kf
= 2
e2 2 0 |b|j$∗ f $ exp(−ikf · x)|a| '(Eb + kf − Ea ) d kf : 2kf0 (2 )3
(228)
Integrating over the photon energy, we obtain (0)
d Wf ;b;a =
kf0 ∗ |i · j (k )|2 d 9f ; 2 f ba f
(229)
where jba (kf ) = b| exp(−ikf · x)|a :
(230)
3.2.2. QED corrections of ?rst order in The QED corrections of ;rst order in are de;ned by the diagrams shown in Fig. 32. Let us consider the derivation of the formulas for the self-energy (SE) corrections (the diagrams (a) – (c)) in detail. Formula (219) gives in the order under consideration (1) (1) 0 Sf ;b;a = '(Eb + kf − Ea ) dE d E gf ;b;a (E ; E) 4b
1 − 2
4b
dE
4a
dE
4a
g(0) (E ; E) f ;b;a
1 2 i
4b
dE
(1) gbb (E)
1 + 2 i
4a
dE
(1) gaa (E)
; (231)
(1) (1) where gaa (E) and gbb (E) are de;ned by the ;rst-order self-energy diagram. Here we have omitted a term of ;rst order in which comes from the factor Z3−1=2 since it has to be combined with the vacuum-polarization (VP) correction. Consider ;rst the diagram (a). According to the Feynman rules, we have 2 i (1; a) 0 0 Gf ((E ; x ); k ; (E; x)) = '(E + k − E) d y d y d z S(E ; x ; y) 0 >(E ; y ; y) 2 i
×
where
>(E ; y ; y) = e
2
i
2
i i S(E ; y; z)A∗f; $ (z)(−2 ie$ ) S(E; z; x) ; 2 2
d ! 0 ( S(E − !; y ; y)) D() (!; y − y)
(232)
(233)
is the kernel of the self-energy operator and D() (!; y − y) is the photon propagator for a non-zero photon mass. According to the de;nition of gf ;b;a (E ; E) (see Eq. (220)), we ;nd a) g(1;f ;b;a (E ; E)
$ ∗ i b|>(E )|nn|e Af; $ |a = 2 n (E − b )(E − n )(E − a )
(234)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 32. The ;rst order QED corrections to the photon emission by a one-electron atom.
and 4b
dE
4a
dE
a) g(1;f ;b;a (E ; E) = −2 i
n=b b|>(b )|nn|e $ A∗f; $ |a n
b − n
+b|> (b )|bb|e $ A∗f; $ |a
;
(235)
where > (b ) ≡ d >()= d |=b . A similar calculation of diagram (b) gives
n=a b|e $ A∗f; $ |nn|>(a )|a (1; b) dE d E gf ;b;a (E ; E) = −2 i a − n 4b 4a n
+b|e $ A∗f; $ |aa|> (a )|a
:
(236)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
175
The second (“reducible”) terms in Eqs. (235) and (236) have to be combined with the second term in Eq. (231). Taking into account that 1 (1) d E gaa (E) = a|> (a )|a ; (237) 2 i 4a 1 2 i
(1)
4b
d E gbb (E) = b|> (b )|b ;
(238)
we obtain
1 1 1 (0) (1) (1) − dE d E gf ;b;a (E ; E) d E gbb (E) + d E gaa (E) 2 4b 2 i 4b 2 i 4a 4a
1 $ ∗ = 2 i b|e Af; $ |a(b|> (b )|b + a|> (a )|a) : 2 For the diagram (c) we ;nd c) g(1;f ;b;a (E ; E) =
1 2 E − b i
i d x d y d z P b (x) 2
∞
−∞
d ! ( S(E − !; x; z)
× eA∗$ (z)$ S(E − !; z; y)) e2 D() (!; x − y) a (y)
1 : E − a
Substituting (240) into (231), we obtain (1; c) dE d E gf ;b;a (E ; E) = −2 i d z eA∗f; $ (z)I$ (b ; a ; z) ; 4b
4a
where I$ (b ; a ; z) = e2
i
2
∞
−∞
d!
(239)
(240)
(241)
d x d y P b (x)( S(b − !; x; z)$ S(a − !; z; y)
× ) D() (!; x − y) a (y) :
(242)
Summing all the ;rst order SE contributions derived above and adding the contribution of the mass-counterterm diagrams (Fig. 33), we ;nd
n=b b|>(b ) − 'm|nn|e $ A∗f; $ |a SE) 0 S(1; = − 2 i '(E + k − E ) a b f f ;b;a b − n n n=a b|e $ A∗f; $ |nn|>(a ) − 'm|a
+
n
a − n
+
d z eA∗f; $ (z)I$ (b ; a ; z)
+ 12 b|e $ A∗f; $ |a(b|> (b )|b
+ a|> (a )|a) :
(243)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 33. The mass-counterterm corrections to the photon emission by a one-electron atom.
A similar calculation of the vacuum-polarization diagrams (Fig. 32, diagrams (d) – (f)) gives
n=b n=a b|UVP |nn|e $ A∗f; $ |a b|e $ A∗f; $ |nn|UVP |a (1; VP) 0 Sf ;b;a = −2 i '(Eb + kf − Ea ) + b − n a − n n n
+
d z eA∗f; $ (z)Q$ (kf0 ; z)
+
(Z3−1=2
−
1)b|e $ A∗f; $ |a
;
(244)
where
∞ 1 dy UVP (x) = d ! Tr [S(!; y; y)0 ] 2 i |x − y| −∞ is the VP potential and $ 0 2 d x d y P b (x)( a (x)D() (k 0 ; x − y) Q (k ; z) = −e i
∞
(245)
d ! Tr [) S(!; y; z)$ S(! + k 0 ; z; y)] : (246) 2 −∞ Some individual terms in Eqs. (243) and (244) contain ultraviolet divergences. These divergences arise solely from the zero- and one-potential terms in the expansion of the electron propagators in powers of the Coulomb potential. Using the standard expressions for the divergent parts of the zero- and one-potential SE terms (see, e.g., [28]) and the Ward identity (Z1 = Z2 ) one easily ;nds that the ultraviolet divergences cancel each other in Eq. (243). As to Eq. (244), the divergent parts incorporate into the charge renormalization factor (e = Z31=2 e0 ). An alternative approach to the renormalization problem consists in using the renormalized ;eld operators R = Z2−1=2 ; AR = Z3−1=2 A, the renormalized electron charge e = e0 + 'e = Z1−1 Z2 Z31=2 e0 , and, respectively, the renormalized Green’s functions from the very beginning. In that approach, additional counterterms arise for the Feynman rules. ×
V.M. Shabaev / Physics Reports 356 (2002) 119–228
177
The vertex and reducible contributions to the SE corrections (third and fourth terms in Eq. (243)) contain infrared divergences which cancel each other when being considered together. In addition to the QED corrections derived in this subsection, we must take into account the contribution originating from changing the photon energy in the zeroth order transition probability (229) due to the QED correction to the energies of the bound states a and b. It follows that the total QED correction to the transition probability of ;rst order in is given by (1) (0)∗ (1) (0) (0) 0 2 d Wf ;b;a = 2 (kf ) 2 Re Gf ;b;a Gf ;b;a d 9f + d Wf ;b;a 0 − d Wf ;b;a 0 : (247) kf =Ea −Eb
kf =a −b
(1; SE) (1; VP) Here G(1) f ;b;a =Gf ;b;a +Gf ;b;a is the QED correction given by Eqs. (243) and (244) in accordance with de;nition (198). Ea ; Eb and a ; b are the energies of the bound states a and b with and without the QED correction, respectively.
3.3. Radiative recombination of an electron with an atom In calculations of processes which contain a free electron in the initial or ;nal or in both states we consider that the interaction with the Coulomb ;eld of the nucleus VC (x) is included ˜ in the source j(x) which leads to a scattering [56] ˜ (i9= − m) (x) = j(x) :
(248)
It means that, after the transition to the “in” operators, the unperturbed Hamiltonian does not contain the interaction with the Coulomb potential. As a result, the Feynman rules contain free-electron propagators, instead of the bound-electron ones, and the vertices corresponding to the interaction of electrons with VC (x) appear. Since we consider the case of a strong Coulomb ;eld, we will sum up in;nite sequences of Feynman diagrams describing the interaction of electrons with the Coulomb potential. As a result of this summation, the free-electron propagators are replaced by the bound-electron propagators, [p0 − H (1 − i0)]−1 = [p0 − H0 (1 − i0)]−1 +[p0 − H0 (1 − i0)]−1 VC [p0 − H0 (1 − i0)]−1 + · · · ;
(249)
where H = H0 + VC and H0 = −i · B + m, and the free-electron wave functions are replaced by the wave functions in the Coulomb ;eld. For instance, the wave function of an incident electron with momentum pi and polarization i is pi i (+)
= Upi i + [pi0 − H0 (1 − i0)]−1 VC Upi i +[pi0 − H0 (1 − i0)]−1 VC [pi0 − H0 (1 − i0)]−1 Upi i + · · · = Upi i + [pi0 − H (1 − i0)]−1 VC Upi i ;
(250)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
where pi =
(pi0 ; pi ),
Upi i =
pi0
=
pi2 + m2 ,
u(pi ; i ) exp(ipi · x) (pi0 =m)(2 )3
(251)
is the free wave function of the incident electron, and u(pi ; i ) is normalized by the condition uu P = 1. Consider the process of the radiative recombination of an electron with momentum pi and polarization i with an (N − 1)-electron atom X (Z−N +1)+ in a state a. As a result of the process, the N -electron atom X (Z−N )+ in a state b and a photon with momentum kf and polarization jf = (0; if ) arise, e− (pi ; i ) + X (Z−N +1)+ (a) → (kf ; jf ) + X (Z−N )+ (b) ;
(252)
where kf =(kf0 ; kf ) and kf0 = |kf |. In this section we will assume that we consider a non-resonant process. It means that the total initial energy (pi0 + Ea ) is not close to any excited-state energy of the N -electron atom. As to the resonance recombination, a detailed description of this process is given in [37] (see also the next sections of the present paper). According to the standard reduction technique [28,56], the amplitude of the process is Sf ;b;ei ;a = b|aout (kf ; jf )b†in (pi ; i )|a = (−iZ3−1=2 )(−iZ2−1=2 )
j$∗ f exp(ikf · y) d4 y d4 z
2kf0 (2 )3
←
×b|Tj$ (y) P (z)|a(− i 9= z − m)
u(pi ; i ) exp(−ipi · z) ; (pi0 =m)(2 )3
(253)
where |a; |b are the vectors of the initial and ;nal states in the Heisenberg representation. Taking into account that ←
P (z)(−i9= − m) = e P (z)A(z) ≡ K(z) P ; z we obtain Sf ;b;ei ;a = 2 '(Eb + ×
0
+
∞
∞
0
− ib|[
kf0
− Ea −
pi0 )(−iZ3−1=2 )(−iZ2−1=2 )
(254)
d y d z A$∗ f (y)
d Gb|j$ (G; y)K(0; P z)|a exp(ikf0 G)
d Gb|K(G; P z)j$ (0; y)|a exp(−ipi0 G) †
(0; z); j$ (0; y)]|a Upi i (z) ;
(255)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
179
where A$f (y) is the wave function of the emitted photon de;ned by Eq. (206) and Upi i (z) is the free wave function of the incident electron de;ned by Eq. (251). To construct the perturbation theory for the amplitude of the process, we introduce the Fourier transform of the corresponding two-time Green’s function, Gf ;ei (E ; E; p0 ; x1 ; : : : ; xN ; x1 ; : : : ; xN −1 )'(E + k 0 − E − p0 )
=
2
i
1 1 N ! (N − 1)!
√
2
∞
−∞
d x0 d x0 exp(iE x0 − iEx0 )
d 4 y d 4 z exp(ik 0 y0 − ip0 z 0 )A$∗ f (y)
×
×0|T (x0 ; x10 ) · · · (x0 ; xN )j$ (y) P (z) P (x0 ; xN −1 ) · · · P (x0 ; x1 )|0 ←
(−i 9= z − m)Upi i (z) :
(256)
As in the derivation of the formulas for the transition amplitudes, we will assume that in zeroth approximation the initial and ;nal states are degenerate in energy with some other states and we will use the same notations for these states as in Section 3.1. As usual, we also assume that a non-zero photon mass is introduced. The spectral representation of Gf ;ei (E ; E; p0 ) can be derived in the same way as for the Green’s function describing the transition amplitude (see Section 3.1 and Appendix E). It gives Gf ;ei (E ; E; p0 ; x1 ; : : : ; xN ; x1 ; : : : ; xN −1 )
=
sb sa 1 1 1 1 1 √ 2 N ! (N − 1)! E − Enb E − Ena na =1 nb =1
0 d y d z A$∗ f (y)0|T (0; x1 ) · · · (0; xN )|nb
×
×
0
+
∞
0
∞
d G nb |j$ (G; y)K(0; P z)|na exp(i(E − Enb + p0 )G)
d G nb |K(G; P z)j$ (0; y)|na exp(i(E − Enb − p0 )G)
−inb |[
†
(0; z); j$ (0; y)]|na na | P (0; xN −1 ) · · · P (0; x1 )|0Upi i (z)
+ terms that are regular functions of E or E if E ≈ Eb(0) and E ≈ Ea(0) :
(257)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
We introduce the Green’s function g(E ; E; p0 ) by gf ;b;ei ;a (E ; E; p0 ) = Pb(0) Gf ;ei (E ; E; p0 )01 · · · 0N −1 Pa(0) :
(258)
From Eq. (257) we have
sb sa ’nb 1 d y d z A$∗ gf ;b;ei ;a (E ; E; p ) = f (y) 2 E − Enb
0
na =1 nb =1
×
+
0
∞
0 ∞
d G nb |j$ (G; y)K(0; P z)|na exp(i(E − Enb + p0 )G)
d G nb |K(G; P z)j$ (0; y)|na exp(i(E − Ena − p0 )G)
− inb |[
†
’†na (0; z); j$ (0; y)]|na Upi i (z) E − Ena
+ terms that are regular functions of E or E if E ≈ Eb(0) and E ≈ Ea(0) ;
(259)
where ’na and ’nb are the wave functions of the initial and ;nal states as de;ned above. Let the contours 4a and 4b surround the poles corresponding to the initial and ;nal levels, respectively, and keep outside other singularities of gf ;b;ei ;a (E ; E; p0 ). This can be performed if the photon mass is chosen as indicated in Section 3.1. Taking into account the biorthogonality condition (71) and comparing (259) with (255) we obtain the desired formula [37] −1=2 0 0 Sf ;b;ei ;a = (Z2 Z3 ) '(Eb + kf − Ea − pi ) dE d E vb† gf ;b;ei ;a (E ; E; pi0 )va ; (260) 4b
4a
where by a we imply one of the initial states and by b one of the ;nal states under consideration. In the case of a single initial state (a) and a single ;nal state (b), it yields −1=2 0 0 Sf ;b;ei ;a = (Z2 Z3 ) '(Eb + kf − Ea − pi ) dE d E gf ;b;ei ;a (E ; E; pi0 ) 4b
1 × 2 i
−1=2
4b
d E gbb (E)
1 2 i
4a
−1=2
4a
d E gaa (E)
:
(261)
For practical calculations by perturbation theory, it is convenient to express the Green’s function gf ;b;ei ;a in terms of the Fourier transform of the (2N − 1)-time Green’s function, gf ;b;ei ;a (E ; E; p0 )'(E + k 0 − E − p0 ) ∞ 1 0 0 = dp10 · · · dpN−1 dp10 · · · dpN0 '(E − p10 · · · − pN−1 )'(E − p10 · · · − pN0 ) N !(N − 1)! −∞ (0)
×Pb Gf ;ei (p10 ; : : : ; pN0 ; k 0 ; p0 ; p10 ; : : : ; pN0 −1 )01 · · · 0N −1 Pa(0) ;
(262)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
181
where Gf ;ei ((p10 ; x1 ); : : : ; (pN0 ; xN ); k 0 ; p0 ; (p10 ; x1 ); : : : ; (pN0 −1 ; xN −1 ))
2 = i
2
1 (2 )2N +1
∞
−∞
d x10 · · · d xN0 −1 d x10 · · · d xN0
×exp(ip10 x10 + · · · + ipN0 xN0 − ip10 x10 − · · · − ipN0 −1 xN0 −1 ) ×
d 4 y d 4 z exp(ik 0 y0 − ip0 z 0 ) A$∗ f (y)
×0|T (x1 ) · · · (xN )j$ (y) P (z) P (xN −1 ) · · · P (x1 )|0 ←
×(−i 9= z − m)Upi i (z) :
(263)
The Green’s function Gf ;ei is constructed using the Wick theorem after the transition in (263) to the interaction representation. Since we have not included the interaction with VC in the unperturbed Hamiltonian, the Feynman rules contain the free-electron propagators and the vertices corresponding to the interaction with VC . Summing over all the insertions of the vertices with VC in the electron lines we replace the free-electron propagators and wave functions by the electron propagators and wave functions in the Coulomb ;eld, according to Eqs. (249) and (250). The free wave function of the incident electron Upi i (x) is replaced by the wave function pi i (+) (x) that can be determined from the equation pi ; i (+)
= Upi i + (pi0 − H0 (1 − i0))−1 VC
pi i (+)
:
(264)
The wave function pi i (+) (x) is a solution of the Dirac equation with the Coulomb potential VC (x) which satis;es the boundary condition [81] p (+) (x)
∼ Up (x) + G + (n)
exp(i|p||x|) ; |x|
|x| → ∞ ;
(265)
where G + (n) is a bispinor depending on n ≡ x= |x|. The apparent expressions for p (+) (x) are given, e.g., in [82]. Thus, we again revert to the Furry picture. The Feynman rules for Gf ;ei diIer from those for Gf only by a replacement of one of the incoming electron propagators (i=2 )S(!; x; y) by the wave function pi i (+) (x). In calculations by perturbation theory a problem appears which is caused by the fact that in zeroth approximation the energy of the (N − 1)-electron atom may coincide with the diIerence of the N -electron energy and the one-electron energy. As a result, some of the terms which are omitted in Eq. (257) are singular functions of E and E if E ≈ Eb(0) and E ≈ Ea(0) . A special analysis of the complete spectral representation of the Green’s function Gf ;ei shows that to eliminate these terms in calculations by formula (260) the following prescription should be used: integrate ;rst over E and keep the point E = Ea(0) + (E − Eb(0) ) outside the contour 4a . In Refs. [45,46] this method was used to derive formulas for the QED corrections to the radiative recombination of an electron with a bare nucleus and for the interelectronic-interaction
182
V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 34. The radiative recombination of an electron with a hydrogen-like atom in zeroth order approximation.
corrections to the radiative recombination of an electron with a high-Z helium-like atom. These formulas are presented in Sections 4.4.1 and 4.4.2. Here we consider another application of the method. 3.4. Radiative recombination of an electron with a high-Z hydrogen-like atom As a practical application of the method, in this section we derive formulas for the radiative recombination of an electron with a high-Z hydrogen-like ion to zeroth and ;rst orders in 1=Z. We will assume that the ;nal state of the helium-like ion is described by a one-determinant wave function 1 ub (x1 ; x2 ) = √ (−1)P Pb1 (x1 ) Pb2 (x2 ) (266) 2 P and the one-electron state |b1 coincides with the initial state |a of the hydrogen-like ion. 3.4.1. Zeroth order approximation To zeroth order, the radiative recombination amplitude is described by the diagram shown in Fig. 34. Formula (261) gives (0) (0) 0 0 Sf ;b;ei ;a = '(Eb + kf − Ea − pi ) dE d E gf ;b;ei ;a (E ; E; pi0 ) ; (267) 4b
4a
where the superscript indicates the order in . According to de;nition (262) and the Feynman rules, we have (E ; E; pi0 )'(E + k 0 − E − pi0 ) g(0) f ;b;ei ;a =
i
2
P
P
(−1)
∞
−∞
dp10 dp20 '(E − p10 − p20 )
0 0 0 ×Pb2 |e $ A$∗ f |p'(p2 + k − pi )
p10
p20
1 − Pb2 + i0
1 '(p10 − E)Pb1 |a ; − a + i0
(268)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
183
where |p ≡ |pi ; i denotes the state vector of the incident electron. We obtain (E ; E; pi0 ) = g(0) f ;b;ei ;a
i
2
E
1 1 b2 |e $ A$∗ : f |p − E − b2 E − a
(269)
Following to the rule for the integration over E and E formulated at the end of Section 3.3, we ;nd (0) dE d E gf ;b;ei ;a (E ; E; pi0 ) = −2 ib2 |e $ A$∗ (270) f |p : 4b
4a
It yields = −2 i'(Eb + kf0 − Ea − pi0 )b2 |e $ A$∗ S(0) f |p f ;b;ei ;a
(271)
or, according to de;nition (198), $∗ G(0) f ;b;ei ;a = −b2 |e $ Af |p :
(272)
The corresponding cross section is d )(0) (2 )4 2 (0) = kf |Gf ;b;ei ;a |2 ; d 9f vi
(273)
where vi is the velocity of the incident electron in the frame of the nucleus. 3.4.2. Interelectronic-interaction corrections of ?rst order in 1/Z The interelectronic-interaction corrections of ;rst order in 1=Z are de;ned by the diagrams shown in Fig. 35. In the order under consideration, formula (261) gives (1) (1) 0 0 Sf ;b;ei ;a = '(Eb + kf − Ea − pi ) dE d E gf ;b;ei ;a (E ; E; pi0 ) 4b
1 − 2
4b
dE
4a
dE
4a
g(0) (E ; E; pi0 ) f ;b;ei ;a
1 2 i
4b
dE
(1) gbb (E)
;
(274)
(1) where gbb (E) is de;ned by the ;rst-order interelectronic-interaction diagram (see Fig. 20). According to de;nition (262) and the Feynman rules, we have
g(1) (E ; E; pi0 )'(E + k 0 − E − pi0 ) f ;b;ei ;a
=
i
2
2 P
P
(−1)
∞
−∞
dp10 dp20 '(E − p10 − p20 )
1 1 1 × 0 0 p1 − Pb1 + i0 p2 − Pb2 + i0 E − a + i0
∞
−∞
d q0 d !
Pb1 Pb2 |I (!)|an'(p20 + ! − q0 )'(p10 − ! − E) × n
184
V.M. Shabaev / Physics Reports 356 (2002) 119–228
Fig. 35. The interelectronic-interaction corrections of ;rst order in 1=Z to the radiative recombination of an electron with a hydrogen-like atom.
×
1 0 0 0 n|e $ A$∗ f |p'(q + k − pi ) q0 − n (1 − i0)
+ Pb1 Pb2 |I (!)|np'(p20 + ! − pi0 )'(p10 − ! − q0 ) ×
1 0 0 n|e $ A$∗ f |a'(q + k − E) q0 − n (1 − i0)
+ Pb2 |e $ A$∗ f |n
q0
1 '(p20 + k 0 − q0 ) − n (1 − i0)
×Pb1 n|I (!)|ap'(q0 + ! − pi0 )'(p10 − ! − E)
1 '(p10 + k 0 − q0 ) − n (1 − i0) ×nPb2 |I (!)|ap'(p20 + ! − pi0 )'(q0 − ! − E) :
+ Pb1 |e $ A$∗ f |n
q0
(275)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
We obtain g(1) (E ; E; pi0 ) = f ;b;ei ;a
i
2
2
1 (−1)P (0) E − E E − a P n
1
b
∞
−∞
185
dp10
1 1 × + p10 − Pb1 + i0 E − p10 − Pb2 + i0 1 n|e $ A$∗ × Pb1 Pb2 |I (p10 − E)|an f |p E − E − n (1 − i0)
+
Pb2 |e $ A$∗ f |n
+
i
2
2
1
E + pi0 − p10 − n (1 − i0)
1 P ( − 1) E − E (0) E − a P n
1
b
Pb1 n|I (p10
∞
−∞
− E)|ap
dp20
1 1 × + p20 − Pb2 + i0 E − p20 − Pb1 + i0 1 n|e $ A$∗ × Pb1 Pb2 |I (pi0 − p20 )|np f |a 0 E − pi − n (1 − i0)
+ Pb1 |e $ A$∗ f |n ×
nPb2 |I (pi0
E+
1
pi0
−
p20
− n (1 − i0)
−
p20 )|ap
:
(276)
The ;rst term in Eq. (276) is conveniently divided into an irreducible (n = Eb(0) − a = b2 ) and a reducible (n = b2 ) part. Since we consider the case of a single ;nal state described by the one-determinant wave function (266) with |b1 = |a, the condition n = b2 means |n = |b2 (otherwise Pb1 Pb2 |I (!)|an = 0). Substituting (276) into (274), integrating over E and E according to the rule formulated at the end of Section 3.3, and using identity (127), we ;nd for the irreducible contribution (1; irred) (1; irred) 0 0 Sf ;b;ei ;a = '(Eb + kf − Ea − pi ) dE d E gf ;b;ei ;a (E ; E; pi0 ) 4b
4a
(−1)P = −2 i'(Eb + kf0 − Ea − pi0 )
×
n=b2 n
P
Pb1 Pb2 |I (Pb1 − a )|an
1 n|e $ A$∗ f |p b2 − n
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
+
n
+
n
+
Pb2 |e $ A$∗ f |n
Pb1 Pb2 |I (pi0 − Pb2 )|np
n
1 Pb1 n|I (Pb1 − a )|ap a + pi0 − Pb1 − n
Pb1 |e $ A$∗ f |n
1 Eb(0)
−
pi0
− n
n|e $ A$∗ f |a
1 0 nPb2 |I (pi − Pb2 )|ap : a + pi0 − Pb2 − n
(277)
For the reducible part we have (1; red) dE d E gf ;b;ei ;a (E ; E; pi0 ) 4b
4a
= 2 i
2
×
2
i
∞
−∞
4b
dp10
dE
P
(−1)P
1 (E − Eb(0) )2
1 1 + 0 0 p1 − Pb1 + i0 E − p1 − Pb2 + i0
×Pb1 Pb2 |I (p10 − a )|ab2 b2 |e $ A$∗ f |p ∞ 1 P =− (−1) d! Pb1 Pb2 |I (! − a )|b1 b2 b2 |e $ A$∗ f |p : (! − Pb1 − i0)2 −∞
(278)
P
The reducible contribution must be considered together with the second term in formula (274). Taking into account that ∞ 1 1 i (1) d E gbb (E) = − 2 dp10 0 b b |I (p10 − b1 )|b1 b2 2 1 2 2 i 4b 2 (p − − i 0) b1 −∞ ∞ 1 − dp10 0 b2 b1 |I (p10 − b1 )|b1 b2 2 (p1 − b2 − i0) −∞
∞ 1 0 0 − dp1 0 b2 b1 |I (p1 − b2 )|b1 b2 ; (279) (p1 − b1 − i0)2 −∞ we ;nd
1 − 2
4b
dE
4a
dE
1 g(0) (E ; E; pi0 ) f ;b;ei ;a 2 i
(1)
4b
d E gbb (E)
∞ 1 b1 b2 |I (!)|b1 b2 = b2 |e $ A$∗ | p 2 d! f 2 (! − i0)2 −∞ ∞ 1 1 − d ! b2 b1 |I (!)|b1 b2 + ; (! − Bb − i0)2 (! + Bb − i0)2 −∞
(280)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
187
where Bb ≡ b2 − b1 . Summing (278) and (280), we obtain for the total reducible contribution ∞ i (1; red) 0 0 1 $∗ d ! b2 b1 |I (!)|b1 b2 Sf ;b;ei ;a = −2 i'(Eb + kf − Ea − pi ) b2 |e $ Af |p 2 2 −∞
1 1 × − : (281) (! + Bb + i0)2 (! + Bb − i0)2 Using the identity 1 1 2 d − =− '(!) (282) (! + i0)2 (! − i0)2 i d! and integrating by parts, we obtain red) = 2 i'(Eb + kf0 − Ea − pi0 ) 12 b2 |e $ A$∗ S(1; f |pb2 b1 |I (Bb )|b1 b2 : f ;b;ei ;a
(283)
In addition to the interelectronic-interaction correction derived in this subsection, we must take into account the contribution originating from changing the photon energy in the zeroth order cross section (273) due to the interelectronic-interaction correction to the energy of the bound state b. It follows that the total interelectronic-interaction correction to the cross section of ;rst order in 1=Z is given by
(0) (0) (2 )4 2 d d )(1) ) ) d (1) = k 2 Re{G(0)∗ − : (284) f ;b;ei ;a Gf ;b;ei ;a } + vi f d 9f k 0 =p0 +a −Eb d 9f k 0 =p0 +a −E (0) d 9f f
i
f
i
b
(1; irred) (1; red) Here G(1) f ;b;ei ;a = Gf ;b;ei ;a + Gf ;b;ei ;a is the interelectronic-interaction correction given by Eqs. (277)
and (283) in accordance with de;nition (198). Eb and Eb(0) are the energies of the bound state b with and without the interelectronic-interaction correction, respectively. 3.5. Photon scattering by an atom
Let us consider the scattering of a photon with momentum ki and polarization ji = (0; ii ) by an atom which is initially in a state a. As a result of the scattering, a photon with momentum kf and polarization jf = (0; if ) is emitted and the atom is left in a state b. In this section we consider the non-resonant photon scattering. This means that the total initial energy of the system (Ea + ki0 ) is not close to any excited-state energy of the atom. The resonant photon scattering will be considered in detail in the next section. According to the standard reduction technique (see, e.g., [28]), the amplitude of the process is Sf ;b;i ;a = b|aout (kf ; f )a†in (ki ; i )|a j$f exp(ikf · y) −1 d4 y d4 z = Disconnected term − Z3 2kf0 (2 )3 j( exp(−iki · z) ×b|Tj$ (y)j( (z)|a i ;
2ki0 (2 )3
(285)
188
V.M. Shabaev / Physics Reports 356 (2002) 119–228
where |a and |b are the vectors of the initial and ;nal states in the Heisenberg representation; kf = (kf0 ; kf ) and ki = (ki0 ; ki ). The ;rst term on the right-hand side of Eq. (285) corresponds to a non-scattering process, i.e., the photon does not interact with the atom. We are interested in the second term which describes the photon scattering by the atom. Let us designate this term by Sscat . f ;b;i ;a To derive the calculation formula for Sscat , in the scattering amplitude we isolate a term f ;b;i ;a which describes the scattering of the photon by the Coulomb potential VC . With this in mind, we write b|Tj(y)j(z)|a = b|[Tj(y)j(z) − 0|Tj(y)j(z)|0]|a + 'ab 0|Tj(y)j(z)|0 :
(286)
The second term on the right-hand side of this equation corresponds to the photon scattering by the Coulomb ;eld and the ;rst term describes the photon scattering by the electrons of the atom. To the ;rst term, only diagrams contribute where the incident photon is connected to at least one external electron line. We denote this term by Scon . We f ;b;i ;a have = −Z3−1 Scon f ;b;i ;a
j$f exp(ikf · y) d4 y d4 z b|[Tj$ (y)j( (z)
2kf0 (2 )3
j( exp(−iki · z) − 0|Tj$ (y)j( (z)|0]|a i
2ki0 (2 )3
= −Z3−1 2 '(Eb
+
kf0
− Ea −
ki0 )
∞
−∞
dt
exp(ikf0 t)
d y d z A$∗ f (y)
b|[Tj$ (t; y)j( (0; z) − 0|Tj$ (t; y)j( (0; z)|0]|aA(i (z) :
(287)
Here the second term is zero if a = b. To construct the perturbation theory for the amplitude of the process, we introduce the Fourier transform of the corresponding two-time Green’s function Gcon (E ; E; k 0 ; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E + k 0 − E − k 0 ) f ;i
=
i
2
2
1 N!
∞
−∞
0
dx dx
0
d 4 y d 4 z exp(iE x0 + ik 0 y0 − iEx0 − ik 0 z 0 )
( 0 0 × A$∗ f (y)Ai (z)0|T (x ; x1 ) · · · (x ; xN )[j$ (y)j( (z)
− 0|Tj$ (y)j( (z)|0] P (x0 ; xN ) · · · P (x0 ; x1 )|0 :
(288)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
189
Introducing t = x0 − z 0 ; t = x0 − z 0 , and G = y0 − z 0 and integrating over z 0 , we obtain Gcon (E ; E; k 0 ; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E + k 0 − E − k 0 ) f ;i
= − '(E + k 0 − E − k 0 ) ×
1 1 2 N !
∞
−∞
d t d t d G exp(iE t + ik 0 G − iEG)
( d y d z A$∗ f (y)Ai (z)0|T (t ; x1 ) · · · (t ; xN )[j$ (G; y)j( (0; z)
− 0|Tj$ (G; y)j( (0; z)|0] P (t; xN ) · · · P (t; x1 )|0 :
(289)
Integrating over t and t we ;nd Gcon (E ; E; k 0 ; x1 ; : : : ; xN ; x1 ; : : : ; xN ) f ;i ∞ 1 1 i i d G exp(ik 0 G) =− 2 N ! n; m E − En + i0 E − Em + i0 −∞ ×
( d y d z A$∗ f (y)Ai (z){.(G) exp[iG(E − En )]0|T (0; x1 ) · · · (0; xN )|n
×n|[j$ (G; y)j( (0; z) − 0|Tj$ (G; y)j( (0; z)|0]|m ×m| P (0; xN ) · · · P (0; x1 )|0 + .(−G) exp[iG(Em − E)] ×0|T (0; x1 ) · · · (0; xN )|nn|[j( (0; z)j$ (G; y) − 0|Tj$ (G; y)j( (0; z)|0]|mm| P (0; xN ) · · · P (0; x1 )|0 + · · ·} :
(290)
It can be shown that the terms which are omitted in Eq. (290) are regular functions of E or E if E ≈ Eb and E ≈ Ea . As in the previous sections of this paper, we assume that in the zeroth approximation the initial and ;nal states of the atom are degenerate in energy with some other states and use the same notations for these states as above. As usual, we also assume that a non-zero photon mass is introduced. We de;ne the Green’s function gcon (E ; E; k 0 ) by f ;b;i ;a (E ; E; k 0 )01 · · · 0N Pa(0) : gf ;b;i ;a (E ; E; k 0 ) = Pb(0) Gcon f ;i From Eq. (290) we have gcon (E ; E; k 0 ) = f ;b;i ;a
1 ’nb 2 n ; n E − Enb a
b
( d y d z A$∗ f (y)Ai (z)
(291)
∞
−∞
d G exp(ik 0 G)
×{.(G) exp[iG(E − Enb )]nb |[j$ (G; y)j( (0; z) − 0|Tj$ (G; y)j( (0; z)|0]|na + .(−G) exp[iG(Ena − E)]
190
V.M. Shabaev / Physics Reports 356 (2002) 119–228
×nb |[j( (0; z)j$ (G; y) − 0|Tj$ (G; y)j( (0; z)|0]|na }
’†na E − Ena
+ terms that are regular functions of E or E if E ≈ Eb(0) and E ≈ Ea(0) :
(292)
Taking into account the biorthogonality condition (71) and comparing (287) with (292), we obtain the desired formula [30] −1 con 0 0 Sf ;b;i ;a = Z3 '(Eb + kf − Ea − ki ) dE d E vb† gcon (E ; E; kf0 )va ; (293) f ;b;i ;a 4b
4a
where we imply by a one of the initial states and by b one of the ;nal states under consideration. In the case of a single initial state (a) and a single ;nal state (b) it yields −1 con 0 0 Sf ;b;i ;a = Z3 '(Eb + kf − Ea − ki ) dE d E gcon (E ; E; kf0 ) f ;b;i ;a 4b
−1=2
1 × 2 i
4b
d E gbb (E)
1 2 i
4a
−1=2
4a
d E gaa (E)
:
(294)
The disconnected term describing the scattering of the photon by the Coulomb ;eld is calculated by the formula j$f exp(ikf · y) j(i exp(−iki · z) −1 discon 4 4 Sf ;b;i ;a = −Z3 'ab d y d z 0|Tj$ (y)j( (z)|0 : (295) 2kf0 (2 )3 2ki0 (2 )3 For practical calculations by perturbation theory it is convenient to express the Green’s function gcon in terms of the Fourier transform of the 2N -time Green’s function, f ;b;i ;a (E ; E; k 0 )'(E + k 0 − E − k 0 ) gcon f ;b;i ;a 1 = N!
∞
−∞
dp10 · · · dpN0 dp10 · · · dpN0 '(E − p10 · · · − pN0 )'(E − p10 · · · − pN0 )
(0)
× Pb Gcon (p10 ; : : : ; pN0 ; k 0 ; k 0 ; p10 ; : : : ; pN0 )01 · · · 0N Pa(0) ; f ;i
where Gcon ((p10 ; x1 ); : : : ; (pN0 ; xN ); k 0 ; k 0 ; (p10 ; x1 ); : : : ; (pN0 ; xN )) f ;i 1 =− (2 )2N
∞
−∞
d x10 · · · d xN0 d x10 · · · d xN0
× exp(ip10 x10 + · · · + ipN0 xN0 − ip10 x10 − · · · − ipN0 xN0 )
(296)
V.M. Shabaev / Physics Reports 356 (2002) 119–228
×
191
d 4 y d 4 z exp(ik 0 y0 − ik 0 z 0 )A$∗ f (y)
×0|T (x1 ) · · · (xN )[j$ (y)j( (z) − 0|Tj$ (y)j( (z)|0] × P (xN ) · · · P (x1 )|0A(i (z) :
(297)
The Green’s function Gcon is constructed using the Wick theorem after the transition in (297) f ;i to the interaction representation. The Feynman rules for Gcon diIer from those for Gf only f ;i by the presence of the incoming photon line which corresponds to the incident photon wave function A(i (x). 3.6. Resonance scattering: spectral line shape The formulas for the scattering amplitudes derived above allow one to perform calculations by perturbation theory in the case where the total initial energy of the system is not close to the energy of an intermediate quasi-stationary state. This is the so-called non-resonant scattering. In the case of resonance scattering, when the initial energy of the system is close to an intermediate-state energy, the direct calculation by perturbation theory according to the formulas derived above leads to some singularities in the scattering amplitude. It means that those formulas cannot be directly applied to the resonance processes. In this section we formulate a method which allows one to calculate the resonance-scattering amplitudes. This method was worked out in Ref. [33]. Another approach, which is limited to the one-electron atom case, was previously developed in Ref. [83]. An attempt to describe a decay process within QED was undertaken in Ref. [84]. The photon scattering by an atom that is initially in its ground state a is now considered for the case of resonance Ea + ki0 ∼ Ed (d = 1; : : : ; s), where Ea is the ground state energy of the atom, ki0 is the incident photon energy, and Ed (d = 1; : : : ; s) are the energies of intermediate atomic states which in zeroth order approximation are equal to the unperturbed energy Ed(0) of a degenerate level. We consider that, as a result of the scattering, a photon of energy kf0 = ki0 is emitted and the atom returns to its ground state a. The calculation of the photon scattering amplitude by the formula derived above leads to a singularity which is caused by the fact that in any ;nite order of perturbation theory one of the energy denominators of an intermediate Green’s function is close to zero. Therefore, in the calculation of the Green’s function gcon (E ; E; k 0 ) we have to go beyond the ;nite-order approximation. With this in f ;a;i ;a mind, let us represent gcon (E ; E; k 0 ) as f ;a;i ;a gcon (E ; E; k 0 ) = f ;a;i ;a
i
2
(E ; k 0 ; E + k 0 ) ga (E )R(−) f
(E ; E; k 0 ) ; + Tgcon f ;a;i ;a
i
2
0 0 gd (E + k 0 )R(+) i (E + k ; k ; E)
i
2
ga (E) (298)
where ga and gd are the Green’s functions de;ned by Eq. (63) with the projectors Pa(0) and (E ; E; k 0 ) is a part of the Green’s function Pd(0) , respectively, k 0 = k 0 + E − E, and Tgcon f ;a;i ;a
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(+) gcon (E ; E; k 0 ) which is regular at E + k 0 ∼ Ed (d = 1; : : : ; s). The operators R(−) f and Ri are f ;a;i ;a constructed by perturbation theory from Eq. (298) which must be considered as their de;nition. Taking into account Eq. (64) and using formula (293), we obtain
Scon = Z3−1 '(kf0 − ki0 )’†a R(−) (Ea ; kf0 ; Ea + ki0 ) f ;a;i ;a f + Z3−1 '(kf0
−
ki0 )
4a
dE
4a
i
2
0 0 gd (Ea + ki0 )R(+) i (Ea + ki ; ki ; Ea )’a
d E va† Tgcon (E ; E; kf0 )va : f ;a;i ;a
(299)
Consider now how the intermediate Green’s function gd (Ea + ki0 ) can be calculated. Let us introduce a quasi-potential Vd (E) by gd (E) = gd(0) (E) + gd(0) (E)Vd (E)gd (E) ;
(300)
where gd(0) = Pd(0) =(E − Ed(0) ). The quasi-potential Vd (E) is constructed by perturbation theory according to Eq. (300) which must be considered as its de;nition. This equation yields Vd (E) = [gd(0) (E)]−1 − [gd (E)]−1 = [gd(0) (E)]−1 − [gd(0) (E) + gd(1) (E) + · · · ]−1 = [gd(0) (E)]−1 gd(1) (E)[gd(0) (E)]−1 + · · · :
(301)
If the quasi-potential Vd (E) is constructed from Eq. (301) to a ;nite order of perturbation theory, the Green’s function gd is determined by gd (E) = [E − Ed(0) − Vd (E)]−1 :
(302)
The Green’s function gd (E) has poles on the second sheet of the Riemann surface, slightly below the right-hand real semiaxis (see Fig. 5), and has no singularities for real E when E ∼ Ed(0) . It means, in particular, that if we take the quasi-potential at least to the lowest order approximation (V (E) ≈ V (Ed(0) )), the Green’s function gd (E) calculated by Eq. (302) has no singularities at E ∼ Ed (d = 1; : : : ; s), and, therefore, the calculation of the resonance-scattering amplitude by Eq. (299) will be correct. The calculation of gd (E) by Eq. (302) eIectively corresponds to summing an in;nite subsequence of Feynman diagrams. For the calculation of gd (E) to the lowest order approximation it is convenient to introduce an operator H by (0)
(0)
H ≡ Ed + Vd (Ed ) :
(303)
The operator H is not Hermitian and has complex eigenvalues. We assume that H is a simple matrix, i.e., its eigenvectors form a complete basis in the space of the unperturbed d-states. We denote its eigenvalues by Ed = Ed − i4d =2, the right eigenvectors by |dR , and the left eigenvectors by |dL . It means H|dR = Ed |dR ;
dL |H = dL |Ed :
(304)
It is convenient to normalize the vectors |dR ; |dL by the condition dL |dR = 'dd :
(305)
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193
Fig. 36. The photon scattering on a one-electron atom in zeroth order approximation.
They satisfy the completeness condition s |dR dL | = I :
(306)
d=1
For gd (E) we obtain gd (E) ≈ (E − H)−1 =
s |dR dL | d=1
E − Ed
:
(307)
In fact, due to T -invariance, Hik = Hki . For this reason the components of the vector dL | can be chosen to be equal to the corresponding components of the vector |dR . In other words, the components of the vector |dR are equal to the complex conjugated components of the vector |dL . If the d states have diIerent quantum numbers, such as the total angular momentum or the parity, one ;nds |dR = |dL ≡ |d. Substituting the lowest order approximation for gd (E) given by Eq. (307) into (299), we obtain in the resonance approximation s (−) (+) a|Rf |dR dL |Ri |a i con 0 0 Sf ;a;i ;a ≈ : (308) '(k − ki ) 2 f E + ki0 − Ed + i4d =2 d=1 a (+) In the resonance approximation, it is suKcient to evaluate the operators R(−) f ; Ri to the lowest order of perturbation theory. They are determined directly from Eq. (298). To demonstrate how the method works we consider below the resonance photon scattering on a one-electron atom. A more general case of a few-electron atom is considered in [33].
3.7. Resonance photon scattering by a one-electron atom In the lowest order the photon scattering by a one-electron atom is described by the diagram shown in Fig. 36. The contribution of this diagram to the Green’s function Gf ;i is ∞ i 2 0 0 d ((E ; x ); k ; k ; (E; x)) = y d z dp0 S(E ; x ; y) e$ '(E + k 0 − p0 ) Gcon f ;i 2 i −∞
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× A$∗ f (y) × A(i (z)
we obtain For gcon f ;a;i ;a gcon (E ; E; k 0 ) = f ;a;i ;a
i
2 i
2
S(p0 ; y; z)
2 e( '(p0 − k 0 − E) i
S(E; z; x) :
(309)
i |aa| 2 e $ A$∗ f 2 E − a i ×
i
2
n
E+
k0
|nn| i |aa| 2 e ( A(i : − n (1 − i0) i 2 E − a
(310)
We consider that Ea + ki0 ∼ Ed and, therefore, represent gcon (E ; E; k 0 ) as the sum of f ;a;i ;a two terms i i (0) 2 gcon (E ; E; k 0 ) = ga(0) (E ) e $ A$∗ g (E + k 0 ) f f ;a;i ;a 2 i 2 d ×
2 2 i (0) i e ( A(i ga (E) + ga(0) (E ) e $ A$∗ f i 2 2 i
n =d
×
n
|nn| e ( A(i ga(0) (E) : E + k 0 − n (1 − i0)
(311)
Comparing this equation with Eq. (298), we derive 2 2 e $ A$∗ R(+) e ( A(i (z) : (312) i (z) = f (y); i i Let us derive now the quasi-potential Vd (E). To the lowest order of perturbation theory it is de;ned by the SE and VP diagrams (see Figs. 12 and 13). As was derived above (see Section 2.5), the contribution of these diagrams to gd (E) is R(−) (y) = f
gd(1) (E) = gd(0) (>SE (E) + UVP )gd(0) :
(313)
Therefore, Vd(1) (E) = [gd(0) (E)]−1 gd(1) (E)[gd(0) (E)]−1 = Pd(0) (>SE (E) + UVP )Pd(0) :
(314)
The operator Pd(0) >SE (E)Pd(0) contains a non-Hermitian part which is responsible for the imaginary part of the energy. The operator (1)
H = d + Vd (a + ki0 )
(315)
acts in the s-dimensional space of the unperturbed states. In reality, due to the fact that the operators >SE and UVP do not mix states with diIerent quantum numbers and among the degenerate one-electron states there are no states with the same quantum numbers, in the case under consideration the right eigenvectors of H coincide with the left eigenvectors, |dR =|dL ≡ |d. However, to keep a general form of the equations, below we will use the right and left
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195
eigenvectors. In the resonance approximation, the amplitude of the process is de;ned by and R(+) Eq. (308), where the operators R(−) i are given by Eq. (312), Ea =a +a|>SE (a )+UVP |a f is the ground state energy including the QED corrections of ;rst order in ; Ed and −4d =2 are the real and imaginary parts of an eigenvalue of H. For the diIerential cross section in the resonance approximation, we obtain
s ( 2 |a|e $ A$∗ f |dR dL |e ( Ai |a| 4 0 0 d ) = (2 ) '(kf − ki ) (Ea + ki0 − Ed )2 + 4d2 =4 d=1 + 2 Re
( ( 2 $∗ ∗ ∗ a|e $ A$∗ f |dR dL |e ( Ai |a| a|e $ Af |dR dL |e ( Ai |a
d¡d
Ea + ki0 − Ed + i4d =2
Ea + ki0 − Ed − i4d =2
d kf :
(316)
For the total cross section, using the optical theorem, we ;nd s ( Re(a|e $ A$∗ i |dR dL |e ( Ai |a)(4d =2) 3 )tot = 2(2 ) (Ea + ki0 − Ed )2 + 4d2 =4 d=1 ( 0 Im(a|e $ A$∗ i |dR dL |e ( Ai |a)(Ed − Ea − ki ) + (Ea + ki0 − Ed )2 + 4d2 =4
:
(317)
Let us discuss, for simplicity, the case s = 2. Only if the d levels have the same quantum numbers, the second term on the right-hand side of Eq. (317) is not equal to zero. In the opposite case, which takes place in the process under consideration, |dR = |dL and, therefore, ( Im(a|e $ A$∗ i |dd|e ( Ai |a) = 0 :
(318)
It follows that the total cross section given by Eq. (317) is the sum of Lorentz-type terms. As to the diIerential cross section, the second term in Eq. (316) is not equal to zero even if the states d = 1; 2 have diIerent quantum numbers. Levels close to each other with identical quantum numbers can appear among doubly excited states of high-Z few-electron atoms [85]. As an example, we can consider the (2s; 2s)0 and (2p1=2 ; 2p1=2 )0 states of a helium-like ion which can arise in the process of recombination of an electron with a hydrogen-like ion. A detailed theory of this process was given in Ref. [37]. The related numerical calculations were presented in Refs. [34,86]. The results obtained in these papers are discussed in Section 4.4.3. 4. Numerical evaluations of QED and interelectronic-interaction corrections in heavy ions 4.1. Methods of numerical evaluations and renormalization procedure In the previous sections we have demonstrated how formulas for the energy shifts and the transition and scattering amplitudes can be derived from the ;rst principles of QED. These formulas usually contain in;nite summations over intermediate electron states (summations over the bound states and integrations over the continuum). These sums are generally evaluated by using analytical expressions for the Dirac–Coulomb Green’s function [4,23,87–91] or by using
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
relativistic ;nite basis set methods [92–97]. In some cases the summation can be performed analytically by employing the generalized virial relations for the Dirac equation [98]. Calculations of most QED corrections require the application of a renormalization procedure. To ;rst order in , one has to renormalize the self-energy and vacuum-polarization diagrams (Figs. 12 and 13). The renormalized expression for the SE correction is given by Eq. (102) which implies using the same covariant regularization for both terms on the right-hand side. For the numerical evaluation of this correction, it is convenient to analytically isolate the ultraviolet divergence in the a|>(a )|a term and cancel it by the counterterm. This can be performed by expanding the Dirac–Coulomb Green’s function in terms of the free Dirac Green’s function, [! − H (1 − i0)]−1 = [! − H0 (1 − i0)]−1 + [! − H0 (1 − i0)]−1 VC [! − H0 (1 − i0)]−1 +[! − H0 (1 − i0)]−1 VC [! − H (1 − i0)]−1 VC [! − H0 (1 − i0)]−1 ; (319) where H0 = · p + m is the free Dirac Hamiltonian. The three terms in Eq. (319) inserted into a|>(a )|a divide the SE correction into zero-, one-, and many-potential terms, respectively.
The ultraviolet divergences in the zero- and one-potential terms and in the counterterm can be cancelled analytically (see Refs. [99 –101] for details). As to the many-potential term, it can easily be shown that it does not contain any ultraviolet divergences. For an overview of other methods of carrying out mass renormalization in numerical calculations, we refer to [4]. The vacuum-polarization correction is determined by Eq. (106) with the VP potential de;ned by (107). Expression (107) is ultraviolet divergent. The simplest way to renormalize this expression is to expand the vacuum loop in powers of the Coulomb potential by employing Eq. (319). According to the Furry theorem, contributions of diagrams with odd numbers of vertices in a vacuum loop (with free Dirac propagators) are equal to zero. Therefore, the ;rst non-zero contribution results from second term in expansion (319). Only this contribution, which is called the Uehling term, is ultraviolet divergent. This term becomes ;nite by charge renormalization. The renormalized expression for the Uehling potential is given by √2 ∞ 2 ∞ 1 t −1 UUehl (r) = − Z d r 4 r ((r ) dt 1 + 2 3 0 2t t2 1 [exp(−2m|r − r |t) − exp(−2m(r + r )t)] ; (320) 4mrt where |e|Z((r) is the density of the nuclear charge distribution ( ((r) d r = 1). The higher order (in VC ) terms are ;nite and their sum is called the Wichmann–Kroll correction [87]. However, the regularization is still required in the second non-zero term due to a spurious gauge dependent piece of the light-by-light scattering contribution. As was shown in [90,91,102], in the calculation of the vacuum polarization charge density based on the partial wave expansion of the Dirac–Coulomb Green’s function, the spurious term does not contribute if the sum over the angular momentum quantum number M is restricted to a ;nite number of terms (|M| 6 K). Thus the Wichmann–Kroll contribution can be calculated by summing up the partial diIerences between the full contribution and the Uehling term. ×
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197
The renormalization procedures described above can be adopted for calculations of the selfenergy and vacuum-polarization screening diagrams [39,40,42,43,103,104] as well as for the QED corrections to the hyper;ne splitting and the bound-electron g factor [6,105 –111]. 4.2. Energy levels in heavy ions 4.2.1. Hydrogen-like ions The relativistic energies of a hydrogen-like ion are determined by the Dirac equation (1). For the point-nucleus case, the Dirac equation can be solved analytically and the binding energy is given by ( Z)2 2 Enj − mc2 = − mc2 ; (321) 2 2 2$ 1 + ( Z=$) + 1 + ( Z=$)2 where $ = n + (j + 1=2)2 − ( Z)2 − (j + 1=2); n is the principal quantum number, and j is the total angular momentum. To obtain the binding energy to higher accuracy, QED and nuclear eIects must be taken into account. The ;nite nuclear size correction is calculated by solving the Dirac equation with the potential of an extended nucleus and by taking the diIerence between the energies for the extended and point nucleus models. This can be performed numerically (see, e.g., [112,113]) or, with a good accuracy, analytically [114]. With a relative accuracy of ∼ 0:2% for Z = 1–100, this correction is given by the following approximate formulas [114]: 2 Z R ( Z)2 2 TEns = mc2 ; (322) [1 + ( Z) fns ( Z)] 2 10n n (˝=mc) 2 ( Z)4 n2 − 1 Z R 2 [1 + ( Z) fnp1=2 ( Z)] 2 mc2 ; TEnp1=2 = 40 n3 n (˝=mc) where = 1 − ( Z)2 ,
(323)
f1s ( Z) = 1:380 − 0:162 Z + 1:612( Z)2 ; f2s ( Z) = 1:508 + 0:215 Z + 1:332( Z)2 ; f2p1=2 ( Z) = 1:615 + 4:319 Z − 9:152( Z)2 + 11:87( Z)3 and R is an eIective radius of the nuclear charge distribution de;ned by 1=2 5 2 3 3 r 4 1 2 r 1 − ( Z) − : R= 3 4 25 r 2 2 7 For the Fermi model of the nuclear charge distribution, N ((r) = ; 1 + exp[(r − c)=a] it is possible to obtain with a very high precision −1 2 a2 3 1+ 2 ; N= 4 c3 c
(324)
(325)
(326)
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
r 2 = 35 c2 + 75 2 a2 ; r 4 = 37 c4 +
18 2 2 2 7 a c
(327) +
31 4 4 7 a
:
(328)
The expectation values for powers of r for a great variety of nuclear charge distribution models are given in Ref. [113]. Next, the QED corrections of ;rst order in should be taken into account. To this order, the QED correction is de;ned by the self-energy and vacuum-polarization diagrams (Figs. 12 and 13). The energy shift from the self-energy diagram (Fig. 12) combined with the related mass counterterm (Fig. 14) is determined by Eq. (102). The self-energy correction for heavy ions was ;rst evaluated by Desederio and Johnson [115] who employed a method suggested by Brown et al. [116]. Later, a more eKcient method was developed by Mohr [89] who calculated this correction to a very high accuracy in a wide interval of Z. The method of the potential expansion of the bound-electron propagator for the calculation of the SE correction to all orders in Z was developed by Snyderman [100] and numerically realized ;rst by Blundell and Snyderman [117]. A very eKcient procedure for the self-energy calculations which is closely related to the methods of Snyderman and Mohr was developed by Yerokhin and Shabaev [101]. An approach in which the ultraviolet divergences are removed by subtractions in coordinate space was worked out by Indelicato and Mohr [118]. The method of the partial-wave renormalization for the calculation of the ;rst order SE correction was developed by Persson et al. [119] and by Quiney and Grant [120]. To date, the most accurate calculations of the SE correction to all orders in Z were performed by Mohr [89,121] and by Indelicato and Mohr [122] for the point nucleus case and by Mohr and SoI [123] for the extended nucleus case. This correction was comprehensively tabulated by Beier and co-workers [124] for ;nite nuclear radii. The highest accuracy for low-Z atoms was gained by Jentschura et al. [125,126]. The VP correction (Fig. 13) is the sum of the Uehling and Wichmann–Kroll contributions. The ;rst contribution can easily be calculated using expression (320) for the Uehling potential. Calculations of the Wichmann–Kroll contribution to all orders in Z were performed ;rst by SoI and Mohr [91] for the extended nucleus case and by Manakov et al. [127] for the point nucleus case. A comprehensive tabulation of this correction for extended nuclei was presented in Ref. [128]. The most accurate calculations for some speci;c ions were accomplished by Persson et al. [129]. The QED corrections of second order in have not yet been calculated completely. Most VP–VP and SE–VP diagrams can be evaluated by the methods developed for the ;rst order SE and VP corrections (see [4,130] and references therein). The most diKcult task consists in the evaluation of the SE–SE contribution. The simplest part of this contribution, the loop-after-loop diagram, was calculated by Mitrushenkov and co-workers [131] and by Mallampalli and Sapirstein [132] for high-Z ions. These numerical calculations were extended to low-Z atoms by Mallampalli and Sapirstein [133] and by Yerokhin [134]. Recently, they were con;rmed by analytical calculations of Yerokhin [135]. As to the residual SE–SE contribution, a speci;c part of it was evaluated by Mallampalli and Sapirstein [132] and an estimate of the complete SE–SE contribution is in progress [136]. For the current status of the corresponding calculations for low-Z atoms, we refer to [137,138].
V.M. Shabaev / Physics Reports 356 (2002) 119–228 Table 1 The ground-state Lamb shift in Point nucleus binding energy Finite nuclear size [39,112] First order SE [123] First order VP [119] Second order QED Nuclear recoil [141] Nuclear polarization [144,145] Lamb shift theory Lamb shift experiment [149]
238
199
U91+ (eV) −132279.92(1) 198.81(38) 355.05 −88.60 ±1:5 0.46 −0.20(10) 465.52(39)±1.5 468(13)
The calculations of the corrections discussed above are based on quantum electrodynamics within the external ;eld approximation. It means that in these calculations the nucleus is considered only as a source of the external Coulomb ;eld VC . The ;rst step beyond this approximation consists in evaluating the nuclear recoil correction. This correction is given by the sum of the lower-order term (194) and the higher-order term (195). For the point nucleus case, an analytical calculation of the lower-order term employing the virial relations for the Dirac equation yields [50] m2 − a2 : (329) 2M The higher order term was numerically evaluated to all orders in Z for point nuclei in Refs. [139,140]. The corresponding calculations for extended nuclei were carried out in Refs. [141,142]. In the case of hydrogen, the highest accuracy was gained in Ref. [143]. Finally, the nuclear polarization correction should be taken into account. This correction results from diagrams describing the interaction of the electron with the nucleus where the intermediate states of the nucleus are excited. It was evaluated by Plunien and SoI [144] and by Ne;odov et al. [145]. In Table 1 we present the individual contributions to the ground-state Lamb shift in 238 U91+ . The uncertainty of the Dirac binding energy results from the uncertainty of the Rydberg constant [146]. As can be deduced from the table, the present status of the experimental precision on the ground-state Lamb shift in hydrogen-like uranium [147–149] provides a test of QED in ;rst order in on the level of about 5%. TEL =
4.2.2. Helium-like ions In heavy helium-like ions, in addition to the one-electron contributions considered in the previous subsection, the two-electron corrections have to be taken into account. To lowest order in , this correction is de;ned by the one-photon exchange diagram (Fig. 20). The calculation of this diagram causes no problem. To second order in , we should account for the two-photon exchange diagrams (Fig. 21), the self-energy screening diagrams (Fig. 22), and the vacuum-polarization screening diagrams (Fig. 23). For the ground state of a helium-like ion, the contribution of the two-photon exchange diagrams is de;ned by Eqs. (134), (139), and (140). The corresponding expressions for the case of a single excited state were obtained in
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Table 2 The two-electron contribution to the ground-state energy in
209
Bi81+ (eV)
One-photon exchange contribution Two-photon exchange within the Breit approximation Two-photon exchange beyond the Breit approximation SE screening VP screening Three- and more photon contribution Total theory [39] Experiment [152]
1897.56(1) −10.64 −0.30(1) −6.73 1.55 0.06(7) 1881.50(7) 1876(14)
Ref. [36]. The derivation of the related formulas for the case of degenerate and quasi-degenerate states by the TTGF method also causes no diKculties. The self-energy and vacuum-polarization screening contributions are given by expressions (153) – (160) (the renormalization of these expressions is considered in detail in Refs. [39,40,42,43]). For the case of quasi-degenerate states, the corresponding formulas are derived by the TTGF method in Refs. [44,75]. The two-photon exchange contribution is conveniently divided into two parts. The ;rst part is the one which can be derived from the Breit equation. For the ground state, it is well determined by the lowest order terms of the Z-expansion series [150], TE (Breit) = 2 [ − 0:15766638 − 0:6356( Z)2 ]m :
(330)
The second part is the remaining one. Evaluated in Refs. [71,73], it was found to be much smaller than the ;rst part. The self-energy and vacuum-polarization screening diagrams were evaluated in Refs. [39,40,103,104]. The related calculations for excited states of helium-like ions were performed for the vacuum-polarization screening diagrams [44] and, in the case of non-mixed states, for the two-photon exchange diagrams [151]. Today, the theoretical uncertainty of the ground-state energy in heavy helium-like ions is completely de;ned by the uncertainty of the one-electron contribution. In this connection, a direct measurement of the two-electron contribution to the ground-state energy in helium-like ions performed in [152] turns out to be rather important. In Table 2 we present the individual two-electron contributions to the ground-state energy in helium-like bismuth. The two-photon exchange contribution is divided into two parts as described above. The three- and more photon contribution is evaluated within the Breit approximation by summing the Z −1 expansion terms for the ground-state energy beginning from Z −3 [39]. For the zeroth order in Z, the coeKcients of this expansion are taken from Ref. [153] and for the second order in Z from Ref. [150]. The uncertainty of this contribution results from QED corrections yet uncalculated. As one can see from the table, to test the second order QED eIects that result from the theory beyond the Breit approximation, the experimental precision has to be improved by an order of magnitude. 4.2.3. Lithium-like ions To date, the highest experimental accuracy was reached in Lamb-shift experiments of lithiumlike ions [154 –157]. In these ions, in addition to the one- and two-electron contributions, the three-electron corrections have to be calculated. To second order in , the three-electron contribution is determined by six diagrams that describe two-photon exchange involving all three
V.M. Shabaev / Physics Reports 356 (2002) 119–228
201
Fig. 37. A typical diagram describing two-photon exchange between three electrons in a lithium-like atom.
electrons. One of these diagrams is shown in Fig. 37. In the case of one electron over the closed (1s)2 shell, the unperturbed wave function is a one-determinant function, 1 u(x1 ; x2 ; x3 ) = √ (−1)P Pa (x1 ) Pb (x2 ) Pv (x3 ) ; (331) 3! P where a and b denote the core states with opposite signs of the angular momentum projection and v indicates the valence state. It can be shown that the derivation of the expressions for the one- and two-electron corrections in three-electron atoms is easily reduced to the derivation in one- and two-electron atoms, respectively. As to the three-electron correction of second order in , it can be derived by the TTGF method using identity (129) and the general rules formulated in Section 2.5.5. Such a derivation for the irreducible part yields [158] IPbPvnQv (Qv − Pv ) − IPanQaQb (Pa − Qa ) TE irred = (−1)P+Q ; (332) Qa + Qb − Pa − n n PQ
where P and Q denote the permutations over the outgoing and incoming electron states, respectively; Iabcd (!) ≡ ab|I (!)|cd. The prime at the sum indicates that terms with vanishing denominator have to be omitted in the summation. The reducible part of the three-electron contribution is [158] 1 TE red = [Ivaav (B)(Iab;ab − Ibv;bv ) + 12 Ibv (333) P 2 Iavvb P ] ; va P (B)Iav;bv P (B)Ivb;va a
where Iab;cd = Iabcd (b − d ) − Ibacd (a − d ); B = v − a ; a is the angular momentum projection of the a electron, b = − a ; vP is the valence electron with the opposite sign of the angular momentum projection. In derivation of Eq. (333) by the TTGF method, some terms containing the vP electron have been cancelled with the corresponding terms from the reducible two-electron contribution of the two-photon exchange diagrams (see Ref. [158] for details). The accurate QED calculations of all the two- and three-electron corrections to the 2p1=2 –2s transition energy up to second order in were performed in Refs. [42,43,158]. Approximate evaluations of these corrections were previously considered in Refs. [159 –164].
202
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Table 3 The 2p1=2 –2s transition energy in
238
U89+ (eV)
One-photon exchange [43] One-electron nuclear size [43] First order QED [119,123] Two-photon exchange within the Breit appoximation [158] Two-photon exchange beyond the Breit approximation [158] Self-energy screening [43] Vacuum polarization screening [42] Three- and more photon exchange [165] Nuclear recoil [139] Nuclear polarization [144,145] One-electron second order QED Total theory Experiment [154]
368.83 −33.35(6) −42.93 −13.54 0.17 1.52 −0.36 0.16(7) −0.07 0.03(1) ±0.20 280.46(9)±0.20 280.59(10)
In Table 3 the individual contributions to the 2p1=2 –2s transition energy in lithium-like uranium are presented. The total theoretical value of the transition energy, 280:46(9) ± 0:20 eV, is in agreement with the related experimental value, 280:59(10) eV [154]. As can be seen from the table, the ;rst-order QED contribution is −42:93 eV while the total second order QED contribution beyond the Breit approximation amounts to 1:33 ± 0:20 eV. Comparing these values with the total theoretical and experimental uncertainties indicates that the present status of the theory for lithium-like uranium provides a test of the QED eIects of ;rst order in on the level of about 0.5% and of the QED eIects of second order in , which result from the theory beyond the Breit approximation, on the level of about 15%. 4.3. Hyper?ne splitting and bound-electron g factor 4.3.1. Hyper?ne splitting in hydrogen-like ions The ground-state hyper;ne splitting of a hydrogen-like ion is conveniently written as [166] 4 m 2I + 1 2 TE = ( Z)3 (334) mc {A( Z)(1 − ')(1 − ) + xrad } : 3 N mp 2I Here mp is the proton mass, is the nuclear magnetic moment, N is the nuclear magneton, and I is the nuclear spin. A( Z) denotes the relativistic factor 1 17 3 A( Z) = (335) = 1 + ( Z)2 + ( Z)4 + · · · ; (2 − 1) 2 8 where = 1 − ( Z)2 . is the nuclear charge distribution correction, is the nuclear magnetization distribution correction (the Bohr–Weisskopf correction), and xrad is the QED correction. The formulas for the ;rst order QED corrections to the hyper;ne splitting are derived in the same way as formulas (115), (117) for the V -SE corrections. For instance, the SE correction is simply determined by Eqs. (115), (117), if V is replaced by the hyper;ne interaction operator |e| ( · [ × r]) Hhfs = (336) 4 r3
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203
Table 4 Theoretical contributions to the ground state hyper;ne splitting in hydrogen-like ions (eV) Ion = N Relativistic value for point nucleus Finite nuclear charge Bohr–Weisskopf eIect QED Total theory [179] Experiment [168–171]
165
Ho66+
4.177(5) 2.326(3)
−0.106(1) −0.020(6) −0.011
2.189(7) 2.1645(5)
185
Re74+
3.1871(3) 3.010
−0.213(2) −0.034(10) −0.015
2.748(10) 2.719(2)
187
Re74+
3.2197(5) 3.041
−0.215(2) −0.035(10) −0.015
2.776(10) 2.745(2)
207
Pb81+
0.592583(9) 1.425
−0.149 −0.053(5) −0.007
1.215(5) 1.2159(2)
209
Bi82+
4.1106(2) 5.839
−0.649(2) −0.061(27) −0.030
5.100(27) 5.0840(8)
and the electron wave functions are replaced by the wave functions of the whole (electron plus nucleus) atomic system. The most complete calculations of the QED and nuclear corrections to the hyper;ne splitting were presented in [107,110] (see also a recent review in [167]). Table 4 shows the individual contributions to the hyper;ne splitting in hydrogen-like ions. The total theoretical values are compared with the experimental results obtained in [168–171]. The uncertainty of the theoretical predictions is mainly determined by the uncertainty of the Bohr–Weisskopf correction which was evaluated within the single-particle nuclear model [107]. This uncertainty should be considered only as an estimate of the order of magnitude of the real error. Except for Ho, the nuclear magnetic moments are taken from Ref. [172]. For Ho the value recommended in [173] is used. In case of 207 Pb, in Ref. [172] two values of the nuclear magnetic moment are given. One ( = 0:592583(9) N ) was measured by the nuclear magnetic resonance (NMR) method [174] while another ( = 0:58219(2) N ) results from an optical pumping (OP) experiment [175]. As was shown in [176], the OP value turns out to be very close to that obtained by NMR if it is corrected for an atomic eIect (see the related discussion in [3]). Therefore, in Table 4 the NMR value for the nuclear magnetic moment of lead is used. Taking into account that the theoretical uncertainties indicated in Table 4 should be considered only as an order of magnitude of the real errors, one can deduce that the total theoretical values are in reasonable agreement with the experimental ones. However, remeasurements of the nuclear magnetic moments by employing modern experimental technique and calculations of the Bohr– Weisskopf eIect within many-particle nuclear models are required to promote investigations of the hyper;ne splitting in hydrogen-like ions. 4.3.2. Hyper?ne splitting in lithium-like ions The energy diIerence between the ground-state hyper;ne splitting components of a lithium-like ion is conveniently written as [38] 1 (2s) 3 m 2I + 1 2 TE(1s)2 2s = ( Z) ] mc [A(2s) ( Z)(1 − '(2s) )(1 − (2s) ) + xrad 6 mp N 2I 1 1 + B( Z) + 2 C( Z) + · · · : (337) Z Z
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Here A(2s) ( Z) denotes the one-electron relativistic factor for the 2s state, 2(1 + )] 2[2(1 + ) + 449 17 A(2s) ( Z) = = 1 + ( Z)2 + ( Z)4 + · · · ; 2 2 (1 + ) (4 − 1) 8 128
(338)
'(2s) is the one-electron nuclear charge distribution correction, (2s) is the one-electron nuclear (2s) is the one-electron QED correction. The terms magnetization distribution correction, and xrad 2 B( Z)=Z and C( Z)=Z describe the interelectronic-interaction corrections of ;rst and second orders in 1=Z, respectively. The ;rst-order interelectronic-interaction correction is conveniently derived using the TTGF method with the closed (1s)2 shell regarded as belonging to the vacuum (see Section 2.5.3). Such a derivation yields [38] FMF FMF CIM TEFMF Ij = jm CIMI jm OIM I MI ; m MI ; m
×
I
(−1)P
c
n =v n
P
+
(−1)P
n
P
+
(−1)P
P
n =c n
P
+
n =v
(−1)P
n =c n
Pv Pc|I (BPcc )|ncn|Hhfs |v v − n
v |Hhfs |nnc|I (BPcc )|PvPc v − n Pv Pc|I (BPv v )|vnn|Hhfs |c c − n c|Hhfs |nv n|I (BPvv )|PvPc c − n
− v |Hhfs |vcv|I (Bvc )|vc
+
c|Hhfs |c v c |I (Bvc )|cv OIMI ;
(339)
c
where F and MF are the total angular momentum of the atom and its projection, v and v are FMF the valence states of electron with quantum numbers (jm) and (j m ), respectively; CIM is the I jm Clebsch–Gordan coeKcient, OIMI is the nuclear wave function, c and c denote the core states, c indicates the angular momentum projection of the core electron, and Bab = a − b . Calculations of the nuclear, QED, and interelectronic-interaction corrections to the hyper;ne splitting in heavy lithium-like ions were performed in Refs. [38,109,167,177–183]. As for hydrogen-like ions, the uncertainty of the theoretical values is mainly determined by the uncertainty of the Bohr–Weisskopf eIect evaluated within the single particle nuclear model. However, in Refs. [109,177] it was found that this uncertainty can be considerably reduced if the experimental value of the 1s hyper;ne splitting in the corresponding hydrogen-like ion is known. The basic
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205
idea of this method is the following. It can be shown that, with a good precision, the ratio of the 2s-Bohr–Weisskopf correction to the 1s-Bohr–Weisskopf correction is a function of the atomic structure only and does not depend on the nuclear structure, (2s) = f( Z) : (1s)
(340)
For Z = 83; f( Z) = 1:078 and, therefore, (2s) = 1:078(1s) . The 1s-Bohr–Weisskopf correction can be found by the equation (1s) =
(1s) (1s) (1s) + TEQED − TEexp TEDirac (1s) TEDirac
;
(341)
(1s) where TEDirac is the relativistic value of the 1s hyper;ne splitting including the nuclear charge (1s) (1s) is the 1s-QED correction, and TEexp is the experimental value distribution correction, TEQED of the 1s hyper;ne splitting. For Z =83, this method predicts the ground-state hyper;ne splitting in lithium-like bismuth to be 0:7971(2) eV [179] (for comparison, the direct evaluation based on the single-particle nuclear model gives 0:800(4) eV). Recently, this value was con;rmed by Sapirstein and Cheng [183] who obtained 0:79715(13) eV. Both theoretical values agree with the experimental one of 0:820(26) eV [155].
4.3.3. Bound-electron g factor The bound-electron g factor in a hydrogen-like ion is de;ned by g(e) = −
(e)
JMJ | z |JMJ
B MJ
;
(342)
where (e) is the operator of the magnetic moment of electron, B is the Bohr magneton, J is the total angular momentum of the electron, and MJ is its projection. For the ground state, a simple relativistic calculation based on the Dirac equation yields [184] g0 = 2 − ( 43 )(1 −
1 − ( Z)2 ) :
(343)
The QED and nuclear eIects give some corrections to this value: g(e) = g0 + TgQED + TgNS :
(344)
Calculation of the nuclear size correction causes no diKculties. For low-Z atoms, this correction can be found by a simple analytical formula [185]. The QED correction of ;rst order in was evaluated without expansion in Z in Refs. [106,108,111] (see also [6]).
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Table 5 The bound-electron g factor in Relativistic value QED Nuclear size correction Total theory [179] Experiment [192]
209
Bi82+
1.7276 0.0029 0.0005 1.7310 1.7343(33)
Direct measurements of the bound-electron g factor in hydrogen-like ions are presently being performed by a GSI-UniversitNat Mainz collaboration [186 –189]. To date, the experimental result obtained for hydrogen-like carbon (C5+ ) [189] amounts to gexp = 2:001041596(5) and agrees with the theoretical predictions of Refs. [6,111], gtheo = 2:001041591(7), and of Ref. [190], gtheo = 2:001041590(2). The collaboration plans to extend these measurements to heavy ions. Another possibility for investigations of the bound-electron g factor was recently proposed in [191]. In this work it was shown that the transition probability between the ground-state hyper;ne splitting components of a hydrogen-like ion, including the ;rst order QED and nuclear corrections, is given by
2 !3 I (n) m (e) w= g − gI ; (345) 3 m2 2I + 1 mp where ! is the transition frequency, g(e) is the bound-electron g factor de;ned above, and gI(n) is the nuclear g factor (both g factors are de;ned to be positive). Formula (345) allows a simple calculation of the QED and nuclear corrections to the transition probability using the corresponding corrections to the bound-electron g factor. In [191] it was found that in the experimentally interesting cases of Pb and Bi, the QED and nuclear corrections increase the transition probability by about 0.3%. In [192] the lifetime of the upper hyper;ne splitting component in 209 Bi82+ was measured to be Gexp =397:5(1:5) s. This result is in good agreement with the theoretical predictions of Ref. [191], Gtheo = 399:01(19) s, and of Ref. [193], Gtheo = 398:89 s. Using formula (345) and the experimental values of the hyper;ne splitting and the transition probability in 209 Bi82+ [168,192], the experimental value of the bound-electron g factor in 209 Bi82+ is found to be 1.7343(33). The corresponding theoretical value is 1.7310. The individual contributions to the bound-electron g factor in 209 Bi82+ are given in Table 5. From this table, it is clear that the QED correction has to be included in order to obtain agreement between theory and experiment. 4.4. Radiative recombination of an electron with a heavy ion In an energetic collision between a high-Z ion and a low-Z target atom, an electron may be captured by the projectile, while a simultaneously emitted photon carries away the excess energy and momentum. This process is denoted as radiative electron capture (REC). Since a loosely bound target electron can be considered as quasi-free, this process is essentially equivalent to radiative recombination (RR) or its time-reversed analogon, the photoelectric eIect. The
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207
relativistic theory of REC was considered in detail in [82,194,195], and the results of this theory are in excellent agreement with experiments [196,197]. A systematic QED theory of the RR process is described in detail in Sections 3.4 and 3.5 of the present paper. In Refs. [34,37,86], this theory was employed to study the process of resonance recombination of an electron with a heavy hydrogen-like ion in the case of resonance with doubly excited (2s; 2s)0 , (2p1=2 ; 2p1=2 )0 ; (2s; 2p1=2 )0; 1 states of the corresponding helium-like ion. Later, this theory was used to evaluate QED corrections to radiative recombination of an electron with a bare nucleus [45] and interelectronic-interaction corrections to radiative recombination of an electron with a heavy helium-like ion [46]. The results of these investigations are brieYy discussed below. More details can be found in the original papers [34,37,45,46,86]. 4.4.1. QED corrections to the radiative recombination of an electron with a bare nucleus We consider the radiative recombination of an electron with momentum pi and polarization i with a bare nucleus that is placed at the origin of the coordinate frame. This corresponds to the projectile system if we study the radiative recombination of a free target electron with a bare heavy projectile. To zeroth order in , the cross section is d )(0) (2 )4 2 (0) 2 = k |G | ; d 9f vi f
(346)
G(0) = −a|e $ A∗f; $ |p = a|e · Af∗ |p ;
(347)
where
|p ≡ |pi ; i is the wave function of the incident electron de;ned by Eq. (264), pi = (pi0 ; pi );
pi0 = pi2 + m2 is the energy of the incident electron, a is the ;nal state of the one-electron atom, kf = (kf0 ; kf ) with kf0 and kf being the photon energy and momentum, respectively, vi is the velocity of the incident electron in the nucleus frame. The QED corrections of ;rst order in are de;ned by diagrams similar to those shown in Fig. 32. The direct calculation by the TTGF method yields for the self-energy correction to the amplitude of the process G(1) SE
=−
n=a a|>(a ) − 'm|nn|e $ A∗f; $ |p n
+
+ 12 a|> (a )|aa|e $ A∗f; $ |p
a|e $ A∗f; $ |nn|>(pi0 ) − 'm|p n
+
a − n
pi0 − n (1 − i0)
d z eA∗f; $ (z)I$ (a ; pi0 ; z)
+
(Z2−1=2
−
1)a|e $ A∗f; $ |p
;
(348)
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where the mass counterterm has been added and ∞ $ 0 2 i I (; p ; z) = e d ! d x d y P a (x)( S( − !; x; z)$ S(p0 − !; z; y) 2 −∞ ×) D() (!; x − y)
pi i (+) (y)
:
(349)
A similar calculation of the VP correction gives
n=a a|UVP |nn|e $ A∗f; $ |p a|e $ A∗f; $ |nn|UVP |p G(1) = − + VP a − n pi0 − n (1 − i0) n n
+
d z eA∗f; $ (z)Q$ (kf0 ; z)
+
(Z3−1=2
−
1)a|e $ A∗f; $ |p
:
(350)
Here UVP (x) is the vacuum-polarization potential de;ned by Eq. (107) and $ 0 2 d x d y P a (x)( pi i (+) (x)D() (k 0 ; x − y) Q (k ; z) = −e ×
i
2
∞
−∞
d ! Tr [) S(!; y; z)$ S(! + k 0 ; z; y)] :
(351)
In addition to these corrections, we have to take into account a contribution originating from changing the photon energy in the zeroth order cross section (346) due to the QED correction to the energy of the bound state a. It follows that the total QED correction of ;rst order in to the cross section is given by
(1) (0) (0) d )QED d ) ) d (2 )4 2 (1) = k 2 Re{G(0)∗ GQED } + − : (352) d 9f vi f d 9f k 0 =p0 −Ea d 9f k 0 =p0 −a f
i
f
i
(1) (1) Here G(1) QED = GSE + GVP is the QED correction given by Eqs. (348) and (350). Ea and a are the energies of the bound state a with and without the QED correction, respectively. Expressions (348) and (350) contain ultraviolet and infrared divergences. While the ultraviolet divergences can be eliminated by the standard renormalization procedure (see the related discussion in Section 3.2.2), the infrared divergences are more diKcult to remove. The infrared-divergent part of G results from the region of small momenta of the virtual photon and is regularized by a non-zero photon mass . An evaluation of this part yields pi2 + m2 − |pi | 2 m 1 : Ginfr = G(0) −log( =m) − log( =m) 1 + 2 log (353) 2 pi 2 2 p + m + |p | i
i
The related contribution to the cross section is 2 + m2 − |p | (0) p 2 i i d )infr d) m ; −2 log( =m) − log( =m) 1 + 2 log = d 9f d 9f pi 2 2 p + m + |p | i
i
(354)
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209
Fig. 38. The radiative recombination accompanied by emission of a soft photon.
where d )(0) = d 9f is the cross section in the zeroth order approximation de;ned by Eq. (346). To cancel the infrared divergent contribution (354), we have to take into account that any experiment has a ;nite energy resolution TE. It means that any numbers of photons of the total energy less than TE can be emitted in the process. It follows that to ;nd the total cross section in the order under consideration, we must include diagrams in which one photon of 0 energy k = k2 + 2 6TE is emitted along with the emission of the photon with the energy kf0 ≈ pi0 − a (we assume TE kf0 ). These diagrams are shown in Fig. 38. Assuming that the energy resolution is suKciently high (TE kf0 ; m), we retain only those contributions from the diagrams shown in Fig. 38 which dominate at TE → 0. Using the standard technique and omitting terms which approach zero at → 0, we obtain for this contribution
d ) (TE) d )(0) m2 = 1 − 2 log 2 − 2 log(TE= ) − 1 + 2 F(|pi |=pi0 ) d 9f d 9f pi 2 + m2 − |p | p 2 i i m 1 ; − 1 + 2 log (355) + log(TE= ) 2 pi 2 2 p + m + |p | i
i
where
F(a) =
0
∞
√ x (1 + a)( x2 + 1 − ax) √ dx 2 log : x +1 (1 − a)( x2 + 1 + ax)
(356)
We can see that infrared divergent parts in Eqs. (354) and (355) cancel each other,
d ) (TE) d )(0) m2 d )infr + = 1 − 2 log 2 − 2 log(TE=m) − 1 + 2 F(|pi |=pi0 ) d 9f d 9f d 9f pi 2 + m2 − |p | p 2 i i 1 m : − 1 + 2 log + log(TE=m) 2 pi 2 2 p + m + |p | i
i
(357)
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Table 6 The relative values of the QED corrections to the total cross section for the radiative recombination into the K-shell of bare uranium [45], expressed in % Impact energy (MeV=u) 100
Correction (1) (1) )en + )bw (1)
)cw
Total 300
(1) (1) + )bw )en (1) )cw
1000
Vacuum polarization (%) 0.126 −0.006
Self energy (%) −0.390
?
0.120
−0.390
0.175
−0.513
−0.003
?
Total
0.173
−0.513
(1) (1) + )bw )en
0.220
−0.591
(1) )cw
0.043
Total
0.263
? −0.591
According to this equation, at a ;xed energy of the incident electron, the QED correction depends on the photon-energy resolution TE and becomes in;nite when TE → 0. It means that the validity of this equation is restricted by the condition ( = )|log(TE=m)|1. For extension of the theory beyond this limit it is necessary to include the radiative corrections of higher orders in (see, e.g., [198,199]). It results in an “exponentiation” of the radiative corrections and removes the singularity for TE → 0. In the derivation of formulas (355) and (357) it has been assumed that the incident electrons have a ;xed energy. These formulas remain also valid in the case when the energy spread of the incident electrons is much smaller than the energy interval TE in which the photons are detected. However, this is not the case for the present REC experiments [196,197], where the energy spread of a quasi-free target electron is much larger than the ;nite photon-energy resolution. In that case, the QED correction to the total RR cross section depends on the form of the energy distribution of the target electron. Since the form of this distribution is not well determined, the only way to study the QED eIects in REC processes is to investigate the cross section into a photon-energy interval which is chosen to be much larger than the eIective energy spread of the quasi-free target electrons and much smaller than the energy of the emitted photon. The Uehling part of the VP correction to the RR cross section and a part of the SE correction were numerically evaluated in Ref. [45]. Expressed in terms of the unperturbed cross section, the individual QED corrections to the total cross section for the radiative recombination into the (1) K-shell of bare uranium are presented in the Table 6. The correction )en results from changing the bound-state energy. It is determined by the term in the square brackets of Eq. (352). The (1) correction )bw corresponds to the irreducible part of the diagrams describing the ;rst order QED (1) results from the diagrams eIect on the bound state electron wave function. The correction )cw describing the QED eIect on the continuum-state wave function.
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211
As in bound-state QED, the Uehling approximation may be expected to account for a dominant part of the VP correction. As to the self-energy correction, we expect that the terms calculated in Ref. [45] give a reasonable estimate of the order of magnitude of the self-energy correction which is beyond the correction depending on the photon-energy interval TE (see Eq. (357)). The relative value of the last correction, which we denote by '(TE), is de;ned by
1 1+ TE '(TE) = −2 + log log ; (358) 1− m where = vi =c. The photon-energy interval has to be chosen in the range 4TE kf0 ; m, where 4 characterizes the energy spread of the incident electrons. In the REC experiments which are performed at GSI, the eIective electron-energy spread is determined by the momentum distribution of the quasi-free target electrons. The width of this spread in the projectile system increases with increasing impact energy. In the case of a N2 gas target, which is presently being employed in the experiments, the eIective energy spread in the projectile (heavy ion) frame amounts to about 10 –40 keV for the impact energy in the range 100 –1000 MeV=u. This value will be considerably reduced in the experiments with a H2 gas target which are under preparation. In the case of a H2 gas target, to satisfy the conditions on TE given above we can choose TE to be 50 keV in the projectile frame for the impact energy 1 GeV=u. The corresponding photon-energy interval in the laboratory (gas-target) frame is determined according to the Lorentz transformation, TEproj = TElab (1 − cos .lab )= 1 − 2 : (359) At a ;xed TEproj ; TElab as a function of the polar angle can be found from this equation. For the photon-energy interval chosen above, '(TE) amounts to −0:59% for an impact energy of 1 GeV=u. Adding the Uehling correction and the part of the SE correction presented in Table 6 to the correction '(TE), we ;nd the QED correction to the total cross section amounts to −0:92% for an impact energy of 1 GeV=u. For a more accurate evaluation of this eIect, complete calculations of all the SE corrections are required. The results of the numerical evaluation of the diIerential cross section can be found in Ref. [45]. Here, we note only that the diIerential cross section at the backward direction vanishes at an impact energy close to 130 MeV=u. In particular, it results in a relatively large contribution of the QED correction to the backward cross section at an energy of 130 MeV=u. At this energy, the QEDen+bw+TE correction is about 0:022 mbarn= sr while the zeroth order cross section amounts only to 0:009 mbarn= sr . 4.4.2. Interelectronic-interaction eCect on the radiative recombination of an electron with a heavy helium-like ion We consider the non-resonant radiative recombination of an electron with momentum pi and polarization i with a heavy helium-like ion in the ground state which is placed at the origin of the coordinate frame. The ;nal state of the system is a lithium-like ion in the state (1s)2 v, where v denotes the valence state. This picture corresponds to the projectile system if we study the radiative recombination of a free target electron with a heavy helium-like projectile. To zeroth order, the amplitude of the process is given by G(0) = v|e · Af∗ |p ;
(360)
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where |v denotes the wave function of the valence electron. The interelectronic-interaction correction of ;rst order in 1=Z can easily be derived using the TTGF method with the closed (1s)2 shell regarded as belonging to the vacuum. Such a derivation yields [46] G(1) int
=
4
Gint l ;
(361)
l=1
where Gint 1
=
c
P
(−1)
n =v
PvPc|I (BPcc )|ncn|e · Af∗ |p
v − n
n
P
1 − cv|I (Bvc )|vcv|e · Af∗ |p ; +
Gint 2 Gint 3
=
c
=
Gint 4 =
(−1)
v|e · Af∗ |nnc|I (BPcc )|PpPc n
P
(−1)
(−1)P
c − kf0 − n (1 − i0)
c|e · Af∗ |nvn|I (BPpv )|PpPc n
P
pi0 − n (1 − i0)
PvPc|I (BpPv )|pnn|e · Af∗ |c n
P
c
P
P
c
(362)
2
c
c + kf0 − n (1 − i0)
;
(363)
;
(364)
:
(365) pi0
= pi2 + m2 is the energy Here |v and |c are the valence and core states, respectively, of the incident electron, kf0 = pi0 − v is the energy of the emitted photon, and c indicates the angular momentum projection of the core electron. Expressions (362) – (365) represent the interelectronic-interaction corrections to the amplitude of the process. The corresponding corrections to the diIerential cross section are d )lint (2 )4 2 = k 2 Re[G(0)∗ Gint (366) l ] : d 9f vi f In addition to this, a contribution originating from a modi;cation of the energy of the emitted photon in the zeroth order cross section due to the interelectronic interaction should be taken into account which is given by int d )en d )(0) d )(0) = − : (367) d 9f d 9f k 0 =p0 −Ev d 9f k 0 =p0 −v f
i
f
i
(1) Here Ev = v +TEint is the energy of the valence electron including the ;rst order interelectronicinteraction correction, (1) TEint = (−1)P PvPc|I (BPvv )|vc : (368) c
P
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213
Table 7 The zeroth order cross section )(0) and the ;rst order interelectronic-interaction correction calculated in Ref. [46], (1) denotes the interelectronic-interaction correction calculated in the screening potential approximation in barns. )scr (1) and )int indicates the results of the rigorous relativistic calculation Impact energy (MeV=u)
)(0)
(1) )scr
(1) )int
2s-state: 100 300 700
41.203 9.105 2.457
−1.393 −0.3345 −0.0979
−2.055 −0.3755 −0.1051
2p1=2 -state: 100 300 700
33.041 5.042 1.065
−2.535 −0.4538 −0.1022
−3.088 −0.3864 −0.0861
2p3=2 -state: 100 300 700
31.489 3.646 0.622
−2.275 −0.3132 −0.0568
−2.896 −0.2804 −0.0489
The total interelectronic-interaction correction to the cross section in ;rst order in 1=Z is 4
(1)
d )int d )int d )lint = en + : d 9f d 9f d 9f
(369)
l=1
int The direct part of the corrections Gint 1 and G2 can be accounted for by a modi;cation of the incoming and outcoming electron wave functions by the screening potential
VC (x) → VC (x) + Vscr (x) ; where
1 Vscr (x) = 2 x
0
x
dy y
2
2 (g1s (y)
(370) +
2 f1s (y))
+
x
∞
2 d y y(g1s (y)
+
2 f1s (y))
:
(371)
Here g1s and f1s are the upper and the lower components of the radial wave function of the ground state, respectively. The screening-potential approximation allows one to account for the dominant part of the interelectronic-interaction eIect and is widely used in practical calculations. Numerical results for the interelectronic-interaction correction to the total cross section of radiative recombination of an electron with helium-like uranium are presented in Table 7 (for more extensive data we refer to [46]). The calculations are carried out for a capture into the 2s; 2p1=2 , and 2p3=2 states of lithium-like uranium and for various projectile energies. The results of the rigorous relativistic treatment are compared with the calculations based on the screening-potential approximation. A deviation of the complete relativistic results from the screening-potential approximation is mainly determined by the term )4int which strongly increases when pi0 comes close to the resonance condition (pi0 − (v − c ) ≈ n ). Numerical results for the diIerential cross section are given in Ref. [46].
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
4.4.3. Resonance recombination of an electron with a heavy hydrogen-like ion We consider the process of recombination of an electron with a very heavy (Z ∼70–110) hydrogen-like ion in its ground state for the case of resonance with doubly excited (2s 2s)0 ; (2p1=2 2p1=2 )0 ; (2s 2p1=2 )0; 1 states of the corresponding helium-like ion. We assume that the experimentally measured quantity is a part of the total cross section which corresponds to the emission of photons with an energy ! ≈ Ed − Er , where Ed (d = 1; 2; 3; 4) are the energies of the doubly excited (2s 2s)0 ; (2p1=2 2p1=2 )0 ; (2s 2p1=2 )0; 1 states, respectively, and Er (r =1; 2; 3; 4) are the energies of singly excited (1s 2p1=2 )0; 1 ; (1s 2s)0; 1 states. The amplitude of this process is given by the sum of the amplitudes of the dielectronic recombination (DR) and radiative recombination (RR) processes (Z−2)+ ∗∗ X (d) → X (Z−2)+ (r)∗ + (!) → · · · ; − 0 (Z−1)+ e (pi ) + X (1s) → X (Z−2)+ (r)∗ + (!) → · · · : Here pi0 is the energy of the incident electron. Among the doubly excited states {d}, there are states with identical quantum numbers ((2s 2s)0 ; (2p1=2 2p1=2 )0 ), while all the singly excited states {r } have diIerent quantum numbers. Main channels of the decay of (1s 2s)1 ; (1s 2p1=2 )1 and (1s 2s)0 ; (1s 2p1=2 )0 states are one- and two-photon transitions to the ground state, respectively. Therefore, two- and three-photon processes give the dominant contribution to the cross section. The cross section of the process can be derived using the method described in Section 3.6. In the resonance approximation, such a derivation yields (see Ref. [37] for details) 2 Wdd |dL |Iˆ|iJ |2 1 2 )(pi0 ) = 2(j + 1) |pi |2 (E1s + pi0 − Ed )2 + 4d2 =4 jlJM
d
Wdd dL |Iˆ|iJ dL |Iˆ|iJ ∗ (E1s + pi0 − Ed + i4d =2)(E1s + pi0 − Ed − i4d =2) d¡d
+ 2 Re
+ WiJ iJ + 2 Re
d
WdiJ dL |Iˆ|iJ E1s + pi0 − Ed + i4d =2
;
(372)
|JMnj lpi0 jl;
where |iJ ≡ (nj l ) = (1 1=2 0) are the quantum numbers of the 1s state of the hydrogen-like ion, Iˆ ≡ I (2s − 1s ), (r) (r) (r) Wdd ; WdiJ = WdiJ ; WiJ iJ = WiJ iJ ; (373) Wdd = r
(r) 2 Wdd = 2 !
Wdi(r)J = 2 !2 Wi(r) J iJ
= 2 !
2
i
r
r
d9f r |Rˆ |dR r |Rˆ |dR ∗ ;
(374)
d 9f r |Rˆ |dR r |Rˆ |iJ ∗ ;
(375)
d 9f |r |Rˆ |iJ |2 ;
(376)
i
i
V.M. Shabaev / Physics Reports 356 (2002) 119–228 2 Rˆ = − e · Af∗ (xn ) :
215
(377)
n=1
% The cross section (372) consists of four terms, ) = 4l=1 )l . The ;rst two terms ()1 ; )2 ) correspond to the DR process, the third term ()3 ) corresponds to the RR process, and )4 describes the interference between the DR and RR processes. The term )2 is caused by the interference of the DR amplitudes on the levels with identical quantum numbers (d; d =1; 2). The magnitude of this term is determined by the overlap of the levels d; d and can be characterized by the non-orthogonality integral dR |dR which is connected with Wdd by the identity (378) Wdd = i(Ed − Ed∗ )dR |dR : Formula (372) was used in [34] for the numerical calculation of the cross section of the resonance recombination of an electron with hydrogen-like uranium. Later, a more accurate calculation was performed by Yerokhin [200]. According to these calculations, the ratio )2 =) which characterizes the overlap eIect amounts up to 30% in the region between the maxima of the curve )(pi0 ). For the parameters of the overlapping levels it was found [34] |dR |dR | = 0:180; |Wdd |= |Ed − Ed | = 0:183 : (379) Using (378) and the identity †
Wdi = ii|(Iˆ − Iˆ )|dR ; expression (372) can be transformed to the following (see Ref. [37] for details) 2 2 1 4d Re(iJ |Iˆ|dR dL |Iˆ|iJ ) 0 )(pi ) = 2(2j + 1) |pi |2 (E1s + pi0 − Ed )2 + 4d2 =4 jlJM d
(380)
Im(iJ |Iˆ† |dR dL |Iˆ|iJ )(E1s + p0 − Ed ) i −2 + WiJ iJ 0 2 2 (E + p − E ) + 4 =4 1s d i d d Im(iJ |(Iˆ − Iˆ† )|dR dL |Iˆ|iJ )(E1s + p0 − Ed )
−2
i
: (381) 0 − E )2 + 42 =4 (E + p 1s d i d d % This expression also consists of four terms ) = 4l=1 )P l . But, in contrast to (372), here the Breit–Wigner part of the cross section is completely contained in the ;rst term ()P 1 ) which is a sum of Lorentz-type terms. The terms )P 2 ; )P 4 again correspond to the interference of the DR amplitudes on the levels d; d = 1; 2 and the interference of the DR and RR processes, respectively. But, unlike )2 ; )4 , the terms )P 2 ; )P 4 do not contain any admixture of Lorentz-type terms. They are given by sums of terms which are odd functions of (E1s + pi0 − Ed ). (We should note that the admixture of Lorentz-type terms in )2 ; )4 is very small and hardly aIects the values )2 =) and )4 =).) The terms )2 ; )4 ()P 2 ; )P 4 ) lead to a deviation of the shape of the individual resonances from the Lorentz shape. This deviation can be characterized by the Low parameter [83] )(Ed − E1s + 4d =2) − )(Ed − E1s − 4d =2) '= : (382) )(Ed − E1s )
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
Calculation of this parameter for d = 2 in the case of recombination with U91+ using the results of Refs. [34,200] yields ' ≈ −0:17. The contributions to ' from the terms )P 2 and )P 4 are equal −0:24 and 0:07, respectively. Summing the interference terms in (381), we obtain a compact formula for ), 1 4d Re (iJ |Iˆ|dR dL |Iˆ|iJ ) 2 2 0 )(pi ) = 2(2j + 1) |pi |2 (E1s + pi0 − Ed )2 + 4d2 =4 jlJM d Im (iJ |Iˆ|dR dL |Iˆ|iJ )(E1s + p0 − Ed ) i
−2
d
(E1s + pi0 − Ed )2 + 4d2 =4
+ WiJ iJ
:
(383)
Formulas (381) and (383) are more convenient for the numerical calculations than formula (372), since they do not require any calculation of the radiative amplitudes. These formulas were employed in Ref. [86] to calculate the resonance recombination of an electron with hydrogen-like lead. 5. Conclusion In the present paper we have considered in detail the two-time Green’s function method for high-Z few-electron systems. This method allows one to formulate a perturbation theory for calculations of various physical quantities in a rigorous and systematic way in the framework of quantum electrodynamics. To demonstrate the eKciency of the method, we have derived formulas for QED and interelectronic-interaction corrections to the energy levels, transition and scattering amplitudes in one-, two-, and three-electron atoms. For the last decade, the TTGF method was intensively employed in calculations of QED eIects in heavy ions. An overview of these calculations was also given in the present paper. Details of the calculations and other applications of the method can be found in Refs. [35 – 46,158]. In particular, in Refs. [42,43] the vacuum-polarization and self-energy screening corrections to the energies of lithium-like ions were calculated. The two-photon exchange diagrams for lithium-like ions were evaluated in Ref. [158]. The second order two-electron diagrams for quasi-degenerate states of helium-like ions are studied in [44,75]. In Ref. [35], the TTGF method was employed to construct an eIective-energy operator for a high-Z few-electron atom. In Ref. [45], this method was used to evaluate the QED corrections to the radiative recombination of an electron with a bare nucleus. The interelectronic-interaction corrections to the radiative electron capture for a helium-like ion were considered in Ref. [46]. In Ref. [80], the interelectronic-interaction corrections to the transition probabilities in helium-like ions are derived. Concluding, the two-time Green’s function method provides a uniform and very eKcient approach for deriving QED and interelectronic-interaction corrections to energy levels, transition probabilities, and cross sections of scattering processes in high-Z few-electron atoms. Using an eIective potential instead of the Coulomb potential of the nucleus allows one to extend this approach to many-electron atoms.
V.M. Shabaev / Physics Reports 356 (2002) 119–228
217
Acknowledgements Many practical calculations by the TTGF method were carried out in collaboration with Anton Artemyev and Vladimir Yerokhin. Stimulating discussions with T. Beier, E.-O. Le Bigot, J. Eichler, P. Indelicato, U. Jentschura, S. Karshenboim, I. Lindgren, P. Mohr, G. Plunien, J. Sapirstein, G. SoI, and S. Zschocke are gratefully acknowledged. Valuable conversations with O. Andreev, D. Arbatsky, I. Bednyakov, M. Sysak, and O. Zherebtsov are also acknowledged. This work was supported in part by RFBR (Grant No. 98-02-18350, Grant No. 98-02-0411 and Grant No. 01-02-17248), by the program “Russian Universities—Basic Research” (Project No. 3930), by DFG (Grant No. 436 RUS 113=616=0-1), and by GSI. Appendix A. QED in the Heisenberg representation In the Heisenberg representation, the basic equations of quantum electrodynamics in the presence of a classical time-independent ;eld A$cl (x) are (i9= − m − eA=cl (x)) (x) = eA=(x) (x) − 'm (x) ; A$ (x) = j$ (x) ; (A.1) P where j$ (x) = (e=2)[ (x)$ ; (x)] is the electron–positron current operator. The state vectors in the Heisenberg representation are time-independent 9t |/ = 0 :
(A.2)
The physical state vectors have to obey a subsidiary condition (9$ A$ (x))(+) |/ = 0 ;
(A.3)
where (9$ A$ (x))(+) is the positive-frequency part of 9$ A$ (x). The Heisenberg operators (x); P (x), and A$ (x) obey the same equal-time permutation relations as the corresponding free-;eld operators. However, in contrast to the free ;elds, the permutation relations for arbitrary times remain unknown. Due to the time-translation invariance, Heisenberg operators obey the following transformation equation: exp(iHt)F(0; x) exp(−iHt) = F(t; x) ;
(A.4)
where H is the Hamiltonian of the system in the Heisenberg representation. For more details, see [21,56,81]. Appendix B. Singularities of the two-time Green’s function in a .nite order of perturbation theory Let us investigate the singularities of the Green’s function G(E) in a ;nite order of perturbation theory. To mth order in e, which corresponds to order m=2 in , it is given by G(m=2) (E; x1 ; : : : ; xN ; x1 ; : : : ; xN )'(E − E ) ∞ 1 1 (−i)m m = d 4 y1 · · · d 4 ym d t d t exp(iE t − iEt) e 2 i N ! −∞ m!
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V.M. Shabaev / Physics Reports 356 (2002) 119–228
× 0|T
in (t
× P in (y1 )(
; x1 ) · · ·
in (t
; xN ) P in (t; xN ) · · · P in (t; x1 )
( in (y1 )Ain (y1 ) · · ·
P (ym )) in
) in (ym )Ain (ym )|0con
;
(B.1)
where the label “con” means that disconnected vacuum–vacuum subdiagrams must be omitted. For simplicity, we omit here the mass renormalization counterterm. The presence of this term would not change the consideration given below. Let us consider the contribution of a diagram of mth order in e. This diagram is de;ned by a certain order of contractions in Eq. (B.1). The contractions between the electron–positron ;elds and between the photon ;elds give propagators (10) and (11), respectively. In this appendix we will use the following representation for these propagators: 0 0 P 0|T in (x) P in (y)|0 = .(x0 − y0 ) n (x) n (y) exp[ − in (x − y )] n ¿0
−.(y0 − x0 )
n ¡0
0|TA(in (x)A)in (y)|0
= −g
()
P
n (x) n (y) exp[
d k exp[ − i
(2 )3
− in (x0 − y0 )] ;
k2 + 2 |x0 − y0 |] exp[ik · (x − y)] : 2 k2 + 2
(B.2)
(B.3)
Here, by de;nition, .(t) = (t + |t |)=(2t) for t = 0 and .(0) = 1=2, and a non-zero photon mass is introduced. Following [13,201], we will use the formalism of time-ordered diagrams [202,203] to investigate the singularities of G(m=2) (E). Let us consider a certain order of the time variables, yi0m ¿ yi0m−1 ¿ · · · ¿ yi0l ¿ t ¿ yi0l−1 ¿ · · · ¿ yi0s ¿ t ¿ yi0s−1 ¿ · · · yi01 which de;nes a time-ordered version of the Feynman diagram. The contribution of the Feynman diagram is the sum of all time-ordered versions. Each time-ordered version is conveniently represented by a diagram in which the vertices are ordered upwards according to increasing time (see, for example, Fig. 39). According to Eqs. (B.2) and (B.3), each electron propagator contains a sum over the whole electron-energy spectrum and each photon propagator contains an integral over the photon momentum. Let us place these sums and integrals in front of the expression for the time-ordered version of the Feynman diagram under consideration. Then, an electron line is characterized by an electron energy n and, according to (B.2), gives a factor (here we are only interested in time-dependent terms) exp[ − in (yi0 − yk0 )] = exp[ − i|n |(yi0 − yk0 )]
for n ¿ 0
and −exp[in (yi0 − yk0 )] = −exp[ − i|n |(yi0 − yk0 )]
for n ¡ 0 ;
where in both cases we consider yi0 ¿ yk0 . A photon line gives a factor exp[ − i k2 + 2 (yi0 − yk0 )] ; where we consider again yi0 ¿ yk0 . Each time yi0 in the diagram is marked by a horizontal dashed line. These lines may intersect other electron and photon lines (see Fig. 39). For the
V.M. Shabaev / Physics Reports 356 (2002) 119–228
219
Fig. 39. A time-ordered version of a Feynman diagram.
point of intersection with an electron line, we introduce a factor exp(i|n |yi0 ) exp(−i|n |yi0 ) = 1, where n is the energy of the intersected electron. For the point of intersection with a photon 0 2 2 line we introduce a factor exp(i k + yi ) exp(−i k2 + 2 yi0 )=1, where k is the momentum of the intersected photon. In addition, we represent the factor exp(iE t ) exp(−iEt) as exp(iE t ) exp(−iEt) = exp[i(E − E)yi0m ] exp[ − i(E − E)yi0m ] · · · ×exp[i(E − E)yi0l ] exp[ − i(E − E)yi0l ] ×exp(iE t ) exp(−iEt ) exp(iEt ) exp(−iEyi0l−1 ) ×exp(iEyi0l−1 ) exp(−iEyi0l−2 ) · · · exp(iEyi0s ) exp(−iEt) :
(B.4)
As a result of all these representations, the integral over times at ;xed intermediate electron states (n) and photon momenta (k) is
Im ≡
∞
−∞
d yi0m
yi0m
−∞
d yi0m−1 : : :
yi0
l
−∞
dt
t
−∞
×exp[i(E − E)yi0m ] exp iE − E −
d yi0l−1 · · ·
|n | −
(m)
×exp iE − E −
(l)
|n | −
yi0s
−∞
dt
(m)
−∞
d yi0s−1 · · ·
yi02
−∞
d yi01
k2 + 2 (yi0m − yi0m−1 ) · · ·
k2 + 2 (yi0l − t )
(l)
t
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×exp iE − ×exp iE −
|n | −
(t )
(t )
|n | −
(s)
k2 + 2 (t − yi0l−1 ) · · ·
k2 + 2 (yi0s − t)
(s)
×exp i− |n | − k2 + 2 (t − yi0s−1 ) · · ·
(t)
(t)
×exp i− |n | − k2 + 2 (yi02 − yi01 ) :
(2)
(B.5)
(2)
% Here (m) |n | denotes the sum of the electron energies from the electron lines which are sand% wiched between the horizontal lines corresponding to the times yi0m and yi0m−1 : (m) k2 + 2 denotes the sum of the photon energies from the photon lines which are sandwiched between the horizontal lines corresponding to the times yi0m and yi0m−1 . Using the identity
0
−∞
d x exp(−i x) =
i
+ i0
;
(B.6)
we easily ;nd Im = 2 '(E − E ) ×
×
E− −
−
%
%
(t ) |n |
%
(t) |n |
−
(m) |n |
i %
−
(t )
i % (t)
−
i %
k2
(m)
+
k2 + 2
k2 + 2 ···
2
+ i0
···
%
−
···
−
E−
(2) |n |
%
(l) |n |
%
−
(s) |n |
i % (2)
−
−
i %
k2 + 2
(l)
i %
k 2 + 2 + i0
(s)
k2 + 2
:
(B.7)
A similar calculation for t ¡ t yields an expression which is obtained from (B.7) by a replacement E → −E in all the denominators. Because each photon line contracts two vertices, at least m=2 denominators in (B.7) have to contain the photon-energy terms and therefore do not contribute to the singularities under consideration. It follows that G(m=2) (E) has isolated poles of all orders till m=2 + 1 at the unperturbed positions of the bound state energies. The separation of these poles from the related cuts is provided by keeping a non-zero photon mass . As to the cuts starting from the lower energy levels, they can be turned down.
V.M. Shabaev / Physics Reports 356 (2002) 119–228
221
Appendix C. Two-time Green’s function in terms of the Fourier transform of the 2N -time Green’s function To prove Eq. (56) we have to show that ∞ 2 1 G(E)'(E − E ) = dp0 · · · dpN0 dp10 · · · dpN0 i N ! −∞ 1 ×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) ×G(p10 ; : : : ; pN0 ; p10 ; : : : ; pN0 ) ;
(C.1)
where the coordinate variables are omitted, for brevity. According to the de;nition of G (see Eq. (12)), Eq. (C.1) is equivalent to ∞ 2 1 G(E)'(E − E ) = dp0 · · · dpN0 dp10 · · · dpN0 i N ! −∞ 1 ×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) ∞ ×(2 )−2N d x10 · · · d xN0 d x10 · · · d xN0 −∞
×exp(ip10 x10 + · · · + ipN0 xN0 − ip10 x10 − · · · − ipN0 xN0 ) ×0|T (x1 ) · · · (xN ) P (xN ) · · · P (x1 )|0 ∞ −2N 2 1 = (2 ) dp0 · · · dpN0 dp20 · · · dpN0 i N ! −∞ 2 ∞ × d x10 · · · d xN0 d x10 · · · d xN0 −∞
×exp[i(E − p20 − · · · − pN0 )x10 + ip20 x20 · · · + ipN0 xN0 ] ×exp[ − i(E − p20 − · · · − pN0 )x10 − ip20 x20 − · · · − ipN0 xN0 ] ×0|T (x1 ) · · · (xN ) P (xN ) · · · P (x1 )|0 :
Using the identity 1 ∞ d ! exp(i!x) = '(x) ; 2 −∞ we obtain
G(E)'(E − E ) = (2 )
−2 2
1 i N!
(C.2)
(C.3)
∞
−∞
d x10 · · · d xN0 d x10 · · · d xN0 '(x10 − x20 ) · · · '(x10 − xN0 )
×'(x10 − x20 ) · · · '(x10 − xN0 ) exp(iE x10 − iEx10 )
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×0|T (x1 ) · · · (xN ) P (xN ) · · · P (x1 )|0 ∞ 1 1 = d x0 d x0 exp(iE x0 − iEx0 ) 2 i N ! −∞ ×0|T (x0 ; x1 ) · · · (x0 ; xN ) P (x0 ; xN ) · · · P (x0 ; x1 )|0 :
(C.4)
The last equation exactly coincides with the de;nition of G(E) given by (18). Appendix D. Matrix elements of the two-time Green’s function between one-determinant wave functions To derive Eq. (61) we use the following two identities. First, if A is a symmetric operator in the coordinates of all electrons, we obtain (see, e.g., [204]) ∗ ∗ Aik ≡ ui |A|uk = (−1)P Pi (71 ) · · · Pi (7N )A(71 ; : : : ; 7N ; 71 ; : : : ; 7N ) 1 N P
×
k1 (71 ) · · ·
kN (7N )
;
(D.1)
where repeated variables {7} imply integration (the integration over x and the summation over ) and A(71 ; : : : ; 7N ; 71 ; : : : ; 7N ) is the kernel of the operator A. Second, if the kernel of the operator A is represented in the form (−1)Q a(7Q1 ; : : : ; 7QN ; 71 ; : : : ; 7N ) ; (D.2) A(71 ; : : : ; 7N ; 71 ; : : : ; 7N ) = Q
we can ;nd
Aik = N ! (−1)P P
∗ Pi1 (71 ) · · ·
∗ PiN (7N )a(71 ; : : : ; 7N ; 71 ; : : : ; 7N ) k1 (71 )
According to (60), we have G(E)01 · · · 0N '(E − E ) =
2 1 i N!
∞
−∞
···
kN (7N )
:
(D.3)
dp10 · · · dpN0 dp10 · · · dpN0
×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 ) 0 0 0 0 ˆ × (−1)PG((p P1 ; 7P1 ); : : : ; (pPN ; 7PN ); (p1 ; 71 ); : : : ; (pN ; 7N )) P
2 1 = (−1)P i N! P
∞
−∞
dp10 · · · dpN0 dp10 · · · dpN0
×'(E − p10 − · · · − pN0 )'(E − p10 − · · · − pN0 )
V.M. Shabaev / Physics Reports 356 (2002) 119–228
223
0 0 0 0 ˆ ×G((p 1 ; 7P1 ); : : : ; (pN ; 7PN ); (p1 ; 71 ); : : : ; (pN ; 7N ))
≡
˜ P1 ; : : : ; 7PN ; 71 ; : : : ; 7N ) : (−1)P G(7
(D.4)
P
Using (D.1) – (D.4) we easily obtain (61). Appendix E. Double spectral representation for the two-time Green’s function describing a transition process Let us consider the function G(E ; E) de;ned as ∞ d t d t exp(iE t − iEt)0|TA(t )B(0)C(t)|0 : G(E ; E) = −∞
(E.1)
Using the transformation rules for the Heisenberg operators and integrating over the time variables, we can derive the following double spectral representation for G(E ; E):
∞ K(W ; W ) L(W ; W ) G(E ; E) = − dW dW + (E − W )(E − W ) (E + W )(E + W ) −∞
∞ M (W ; !) N (W ; !) + dW d! + (E − W )(k 0 + !) (E + W )(k 0 − !) −∞
∞ P(!; W ) Q(!; W ) − d! dW ; (E.2) + (k 0 − !)(E − W ) (k 0 + !)(E + W ) −∞ where k 0 = E − E , '(W − En )'(W − Em )0|A(0)|nn|B(0)|mm|C(0)|0 ; K(W ; W ) =
(E.3)
n; m
L(W ; W ) =
'(W − En )'(W − Em )0|C(0)|nn|B(0)|mm|A(0)|0 ;
(E.4)
n; m
M (W ; !) =
'(W − En )'(! − Em )0|A(0)|nn|C(0)|mm|B(0)|0 ;
(E.5)
'(W − Em )'(! − En )0|B(0)|nn|C(0)|mm|A(0)|0 ;
(E.6)
n; m
N (W ; !) =
n; m
P(!; W ) =
'(! − En )'(W − Em )0|B(0)|nn|A(0)|mm|C(0)|0 ;
(E.7)
n; m
Q(!; W ) =
n; m
'(! − Em )'(W − En )0|C(0)|nn|A(0)|mm|B(0)|0 :
(E.8)
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Physics Reports 356 (2002) 229–365
The mathematical physics of rainbows and glories John A. Adam Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA Received May 2001; editors: J: Eichler ; T:F: Gallagher
Contents 1. Introduction 1.1. Structure and philosophy of the review 1.2. The rainbow: elementary physical features 1.3. The rainbow: elementary mathematical considerations 1.4. Polarization of the rainbow 1.5. The physical basis for the divergence problem 2. Theoretical foundations 2.1. The supernumerary rainbows; a heuristic account of Airy theory 2.2. Mie scattering theory 3. Glories 3.1. The backward glory 3.2. Rainbow glories 3.3. The forward glory 4. Semi-classical and uniform approximation descriptions of scattering 5. The complex angular momentum theory: scalar problem 5.1. The quantum mechanical connection 5.2. The poles of the scattering matrix 5.3. The Debye expansion 5.4. Geometrical optics r7egimes 5.5. Saddle points 5.6. The glory
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5.7. Summary of the CAM theory for rainbows and glories 5.8. A synopsis: di9ractive scattering, tunneling e9ects, shape resonances and Regge trajectories [89] 6. The electromagnetic problem 6.1. Polarization 6.2. Further developments on polarization: Airy theory revisited 6.3. Comparison of theories 6.4. Non-spherical (non-pendant) drops 6.5. Rainbows and glories in atomic, nuclear and particle physics 7. The rainbow as a di9raction catastrophe 8. Summary 8.1. The rainbow according to CAM theory Acknowledgements Appendix A. Classical scattering; the scattering cross section A.1. Semi-classical considerations: a pr7ecis Appendix B. Airy functions and Fock functions Appendix C. The Watson transform and its modiAcation for the CAM method Appendix D. The Chester–Friedman–Ursell (CFU) method References
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[email protected] (J.A. Adam). c 2002 Published by Elsevier Science B.V. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 7 6 - X
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Abstract A detailed qualitative summary of the optical rainbow is provided at several complementary levels of description, including geometrical optics (ray theory), the Airy approximation, Mie scattering theory, the complex angular momentum (CAM) method, and catastrophe theory. The phenomenon known commonly as the glory is also discussed from both physical and mathematical points of view: backward glories, rainbow-glories and forward glories. While both rainbows and glories result from scattering of the incident ◦ radiation, the primary rainbow arises from scattering at about 138 from the forward direction, whereas the (backward) glory is associated with scattering very close to the backward direction. In fact, it is a more complex phenomenon physically than the rainbow, involving a variety of di9erent e9ects (including surface waves) associated with the scattering droplet. Both sets of optical phenomena—rainbows and glories—have their counterparts in atomic, molecular and nuclear scattering, and these are addressed also. The conceptual foundations for understanding rainbows, glories and their associated features range from classical geometrical optics, through quantum mechanics (in particular scattering from a square well potential; the associated Regge poles and scattering amplitude functions) to di9raction catastrophes. Both the scalar and the electromagnetic scattering problems are reviewed, the latter providing details about the polarization of the rainbow that the scalar problem cannot address. The basis for the complex angular momentum (CAM) theory (used in both types of scattering problem) is a modiAcation of the Watson transform, developed by Watson in the early part of this century in the study of radio wave di9raction around the earth. This modiAed Watson transform enables a valuable and accurate approximation to be made to the Mie partial-wave series, which while exact, converges very slowly at high frequencies. The theory and many applications of the CAM method were developed in a fundamental series of papers by Nussenzveig and co-workers (including an important interpretation based on the concept of tunneling), but many other contributions have been made to the understanding of these beautiful phenomena, including descriptions in terms of so-called di9raction catastrophes. The rainbow is a Ane example of an observable event which may be described at many levels of mathematical sophistication using distinct mathematical approaches, and in so doing the connections between several seemingly unrelated areas within physics c 2002 Published by Elsevier Science B.V. become evident. PACS: 42.15.−i; 42.25.−p; 42.25.FX; 42.68.Mj Keywords: Rainbow; Glory; Mie theory; Scattering; Complex angular momentum; Di9raction catastrophe
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Like the appearance of a rainbow in the clouds on a rainy day, so was the radiance around him. This was the appearance of the likeness of the glory of the Lord... The Book of Ezekiel, chapter 1, verse 28 (New International Version of the Bible).
1. Introduction 1.1. Structure and philosophy of the review The rainbow is at one and the same time one of the most beautiful visual displays in nature and, in a sense, an intangible phenomenon. It is illusory in that it is not of course a solid arch, but like mirages, it is nonetheless real. It can be seen and photographed, and described as a phenomenon of mathematical physics, but it cannot be located at a speciAc place, only in a particular direction. What then is a rainbow? Since many levels of description will be encountered along the way, the answers to this question will take us on a rather long but fascinating journey in the footsteps of those who have made signiAcant contributions to the subject of “light scattering by small particles”. Let us Arst ‘listen’ to what others have written about rainbows and the mathematical tools with which to understand them. “Rainbows have long been a source of inspiration both for those who would prefer to treat them impressionistically or mathematically. The attraction to this phenomenon of D7escartes, Newton, and Young, among others, has resulted in the formulation and testing of some of the most fundamental principles of mathematical physics.” K. Sassen [1]. “The rainbow is a bridge between two cultures: poets and scientists alike have long been challenged to describe it: : : Some of the most powerful tools of mathematical physics were devised explicitly to deal with the problem of the rainbow and with closely related problems. Indeed, the rainbow has served as a touchstone for testing theories of optics. With the more successful of those theories it is now possible to describe the rainbow mathematically, that is, to predict the distribution of light in the sky. The same methods can also be applied to related phenomena, such as the bright ring of color called the glory, and even to other kinds of rainbows, such as atomic and nuclear ones.” H.M. Nussenzveig [2]. “Probably no mathematical structure is richer, in terms of the variety of physical situations to which it can be applied, than the equations and techniques that constitute wave theory. Eigenvalues and eigenfunctions, Hilbert spaces and abstract quantum mechanics, numerical Fourier analysis, the wave equations of Helmholtz (optics, sound, radio), SchrRodinger (electrons in matter), Dirac (fast electrons) and Klein–Gordon (mesons), variational methods, scattering theory, asymptotic evaluation of integrals (ship waves, tidal waves, radio waves around the earth, di9raction of light)—examples such as these jostle together to prove the proposition.” M.V. Berry [3]. The three quotations above provide a succinct yet comprehensive survey of the topic addressed in this review: the mathematical physics of rainbows and glories. An attempt has been made to provide complementary levels of description of the rainbow and related phenomena; this mirrors
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to some extent the historical development of the subject, but at a deeper level it addresses the fact that, in order to understand a given phenomenon as fully as possible, it is necessary to study it at as many complementary levels of description as possible. In the present context, this means both descriptive and mathematical accounts of the rainbow and related phenomena, the latter account forming the basis of the paper: it is subdivided into the various approaches and levels of mathematical sophistication that have characterized the subject from the investigations of D7escartes down to the present era. There are several classic books and important papers that have been drawn on frequently throughout this paper: the book by Greenler on rainbows, halos and glories [4] has proved invaluable for the descriptive physics in this introduction; the article by Nussenzveig [2] from which the second quotation is taken is an excellent introduction to both the physics and the qualitative description of the various mathematical theories that exist for the rainbow. The two papers [5,6] by the same author constitute a major thread running throughout this article, but particularly so in Section 5 (complex angular momentum theory). Van de Hulst’s book on light scattering by small particles [7] is a classic in the Aeld, and for that reason is often cited, both in this article and in many of the references. Indeed, to quote from Section 13:2 in that book “The rainbow is one of the most beautiful phenomena in nature. It has inspired art and mythology in all people, and it has been a pleasure and challenge to the mathematical physicists of four centuries. A person browsing through the old literature receives the impression that a certain a9ection for this problem pervades even the driest computations.” The very next sentence in the above citation is most aS propos, since it also accurately reTects this author’s hopes for the present article “The writer hopes that the following report, though it has to be concise and must leave out most of the history, will to some extent demonstrate this mathematical beauty.” Another important book, less comprehensive and mathematically complete than [7], but nonetheless extremely valuable from both physical and mathematical viewpoints, is that by Tricker on meteorological optics [8]. The book by Bohren and Hu9man [9] provides a great deal of information on the absorption and scattering of light by small particles and diverse applications, with many references. At the time of writing, the most comprehensive and up-to-date book on di9raction e9ects in semi-classical scattering theory and its applications is that by Nussenzveig [10]. It is a seminal work, and incorporates many of the topics addressed here (amongst others), and should be consulted by anyone wishing to study the subject from an expert in the Aeld. No attempt is made here to discuss the historical development of the theory of the rainbow except insofar as it is germane to the context; that topic is superbly treated in the book by Boyer [11]. A noteworthy and somewhat technical account of more recent historical developments can be found in the review article by Logan [12]. It is a survey of early studies in the scattering of plane waves by a sphere. This is extremely important and interesting from a historical perspective: Logan provides 103 references, and in so doing notes that the literature appears to be characterized by writers who, it seems, failed to recognize the signiAcance of the contributions made by their predecessors and contemporaries. There are several instances of the “rediscovery of the wheel” (so to speak), and the author has performed a valuable task in identifying the “lost” contributions to the subject.
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In the rest of this introduction, elementary physical and mathematical descriptions (e.g. geometrical optics) of the rainbow are provided. Section 2 addresses the capacity of Airy’s theory to account for, among other things, the primary bow and its associated supernumeraries. In Section 3 glories (backward, forward and rainbow-modiAed) are described; Section 4 addresses the semiclassical description of atomic, molecular and nuclear rainbows and glories, as well as di9ractive=tunneling e9ects in these areas and in particle physics. Section 5 is devoted to a summary of the complex angular momentum (CAM) theory for the scalar scattering problem; a fascinating and very powerful tool for understanding many of the subtleties of wave scattering by both impenetrable and transparent (or even absorbing) spheres. It is appropriate to include the topic of the glory again in this section, because much of the CAM theory developed by Nussenzveig addressed both rainbows and glories. In Section 6 further developments are discussed, including the full electromagnetic problem (Mie scattering theory), polarization, alternative models, the e9ects of non-spherical droplets, and a comparison of theories. A brief account of rainbows and glories in atomic, molecular and nuclear scattering is also provided. Section 7 contains an account of the relevant aspects of ‘di9raction catastrophe’ scattering to the problem at hand. The summary in Section 8 precedes four appendices: Appendix A summarizes aspects of classical scattering relevant to the present article, and previews rainbow and glory scattering, forward peaking and orbiting, all of which reappear in a semiclassical context. Appendix B provides brief details of the Airy and Fock functions referred to in the article. Appendix C contains a brief account of the Watson transform and its modiAcation to become the basis of the CAM theory discussed earlier, while Appendix D is a brief summary of the Chester–Friedmann–Ursell (CFU) method. 1.2. The rainbow: elementary physical features Let us return to the question asked above: what then is a rainbow? It is sunlight, displaced by reTection and dispersed by refraction in raindrops. It is seen by an observer with his or her back to the sun (under appropriate circumstances). As shown in Fig. 1(a), the primary rainbow, which is the lowest and brightest of two that may be seen, is formed from two refractions and one reTection in myriads of raindrops (the path for the secondary rainbow is shown in Fig. 1(b); Fig. 1(c) illustrates notation referred to in the text). For our purposes we may consider the path of a ray of light through a single drop of rain, for the geometry is the same for all such drops and a given observer. Furthermore, we can appreciate most of the common features of the rainbow by using the ray theory of light; the wave theory of light is needed to discuss the Aner features such as the supernumerary bows (discussed below). The Arst satisfactory explanation for the existence and shape of the rainbow was given by Ren7e D7escartes in 1637 (he was unable to account for the colors however; it was not until thirty years later that Newton remedied this situation). D7escartes used a combination of experiment and theory to deduce that both the primary and secondary bow (larger in angular diameter and fainter than the primary) are caused by refraction and reTection in spherical raindrops. He correctly surmised that he could reproduce these features by passing light through a large water-Alled Task—a really big “raindrop”. Since the laws of refraction and reTection had been formulated some 16 years before the publication of D7escartes’ treatise by the Dutch scientist Snell, D7escartes could calculate and trace the fate of parallel rays from the sun impinging on
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Fig. 1. (a) the basic geometry of ray paths in a spherical raindrop for the primary rainbow; O is the center of the drop, and i, r, respectively, are the angles of incidence and refraction. The incident and exiting ‘wavefronts’ A A and B B are also shown (refer to Section 2.1), (b) the corresponding geometry for the formation of the secondary rainbow, (c) the geometry of an incident ray (1) showing the externally reTected ray (1 ), the transmitted ray (2) and externally refracted ray (2 ), etc. All externally transmitted rays are denoted by primed numbers.
a spherical raindrop. As can be seen from Fig. 2, such rays exit the drop having been deviated from their original direction by varying but large amounts. The ray along the central axis (#1) ◦ will be deviated by exactly 180 , whereas above this point of entry the angle of deviation ◦ decreases until a minimum value of about 138 occurs (for yellow light; other colors have slightly di9erent minimum deviation angles). For rays impinging still higher above the axis the
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Fig. 2. The paths of several rays through a spherical raindrop illustrating the di9erences in total deTection angle for di9erent angles of incidence (or equivalently, di9erent impact parameters). The ray #7 is called the rainbow ray and this ray deAnes the minimum angle of deTection in the primary rainbow. (Redrawn from [4].)
deviation angle increases again. The ray of minimum deviation (#7 in Fig. 2) is called the rainbow ray. The signiAcant feature of this geometrical system is that the rays leaving the drop are not uniformly spaced: those “near” the minimum deviation angle are concentrated around it, whereas those deviated by larger angles are spaced more widely (see also Fig. 3). Put di9erently, in a small (say half a degree) angle on either side of the rainbow angle ( ◦ 138 ) there are more rays emerging than in any other one degree interval. It is this concentration of rays that gives rise to the (primary) rainbow, at least as far as its light intensity is concerned. In this sense it is similar to a caustic formed on the surface of the tea in a cup when appropriately illuminated. The rainbow seen by any given observer consists of those deviated rays that of ◦ course enter his eye. These are those that are deviated by about 138 from their original direction (for the primary rainbow). Thus the rainbow can be seen by looking in any direction that is ◦ about 42 away from the line joining one’s eye to the shadow of one’s head (the antisolar ◦ point); the 42 angle is supplementary to the rainbow angle. This criterion deAnes a circular arc (or a full circle if the observer is above the raincloud) around the antisolar point and hence all raindrops at that angle will contribute to one’s primary rainbow. Of course, on level ground, at most a semi-circular arc will be seen (i.e. if the sun is close to setting or has just risen), and usually it will be less than that: full circular rainbows can be seen from time to time at high altitudes on land or from aircraft. In summary, the primary rainbow is formed by the deTected rays from all the raindrops that lie on the surface of a cone with vertex (or apex) at the eye, ◦ axis along the antisolar direction and semi-vertex angle of 42 . The same statement holds for the ◦ ◦ secondary rainbow if the semi-vertex angle is about 51 (the supplement of a 129 deviation).
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Fig. 3. A more detailed version of Fig. 2 showing the “Airy wavefront” (which is perpendicular to all rays of class 3’ in Fig. 1(c)). Also shown, following Nussenzveig [2] are the caustics (one real, one imaginary) of these rays; these are envelopes of the ray system. (Redrawn from [2].)
These cones will be di9erent for each observer, so each person has his or her own personal rainbow. Up to this point, we have been describing a generic, colorless type of rainbow. Blue and violet light get refracted more than red light: the actual amount depends on the index of refraction of the raindrop, and the calculations thereof vary slightly in the literature because the wavelengths W (Angstrom chosen for “red” and “violet” may di9er slightly. Thus, for a wavelength of 6563 A ◦ −10 W = 10 W units; 1 A m) the cone semi-angle is about 42:3 , whereas for violet light of 4047 A ◦ ◦ wavelength, the cone semi-angle is about 40:6 , about a 1:7 angular spread for the primary bow. A similar spread (dispersion) occurs for the secondary bow, but the additional reTection reverses the sequence of colors, so the red in this bow is on the inside of the arc. In principle more than two internal reTections may take place inside each raindrop, so higher-order rainbows (tertiary, quaternary, etc.) are possible. It is possible to derive the angular size of such a rainbow after any given number of reTections (Newton was the Arst to do this). Newton’s contemporary, Edmund Halley found that the third rainbow arc should appear as a ◦ circle of angular radius about 40 around the sun itself. The fact that the sky background is so bright in this vicinity, coupled with the intrinsic faintness of the bow itself would make such a bow almost impossible, if not impossible to see (but see [13]). Jearl Walker has used a laser beam to illuminate a single drop of water and traced rainbows up to the 13th order, their positions agreeing closely with predictions. Others have traced 19 rainbows under similar laboratory conditions [14].
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The sky below the primary rainbow is often noticeably brighter than the sky outside it; indeed the region between the primary and secondary bow is called Alexander’s dark band (after Alexander of Aphrodisias who studied it in connection with Aristotle’s (incorrect) theory of the rainbow). Raindrops scatter incident sunlight in essentially all directions, but as we have seen, the rainbow is a consequence of a “caustic” or concentration of such scattered light in a particular region of the sky. The reason the inside of the primary bow (i.e. inside the cone) is bright is that all the raindrops in the interior of the cone reTect light to the eye also, (some occurring from direct reTections at their surfaces) but it is not as intense as the rainbow light, and it is composed of many colors intermixed. Similarly, outside the secondary bow a similar (but less obvious) e9ect occurs (see Figs. 4:3(b) and 4:7 in [15]). Much of the scattered light then, comes from raindrops through which sunlight is refracted and reTected: these rays do not emerge between ◦ ◦ the 42 and 51 angle. This dark angular band is not completely dark, of course, because the surfaces of raindrops reTect light into it; the reduction of intensity, however, is certainly noticeable. Another commonly observed feature of the rainbow is that when the sun is near the horizon, the nearly vertical arcs of the rainbow near the ground are often brighter than the upper part of the arc. The reason for this appears to be the presence of drops with varying sizes. Drops smaller than, say 0.2–0:3 mm (about 1=100 in) are spherical: surface tension is quite suYcient to keep the distorting e9ects of aerodynamical forces at bay. Larger drops become more oblate in shape, maintaining a circular cross section horizontally, but not vertically. They can contribute signiAcantly to the intensity of the rainbow because of their size (≈ 1=25 in, i.e. ≈ 1 mm or larger) but can only do so when they are “low” on the cone, for the light is scattered in a horizontal plane in the exact way it should to produce a rainbow. These drops do not contribute signiAcantly near the top of the arc because of their non-circular cross section for scattering. This will be discussed in more detail in Section 6.4. Small drops, on the other hand contribute to all portions of the rainbow. Drop size, as implied above, can make a considerable di9erence to the intensity and color of the rainbow. The “best” bows are formed when the drop diameter is &1 mm; as the size decreases the coloration and general deAnition of the rainbow becomes poorer. Ultimately, when the drops are about 0:05 mm or smaller in diameter, a broad, faint, white arc called a fogbow occurs. When sunlight passes through these very tiny droplets the phenomenon of di9raction becomes important. Essentially, due to the wave nature of light, interactions of light with objects comparable to, (or not too much larger than) a typical wavelength, a light beam will spread out. Thus the rainbow colors are broadened and overlap, giving rise in extreme cases to a broad white fogbow (or cloudbow, since droplets in those typically produce such bows). As Greenler points out, these white rainbows may sometimes be noted while Tying above a smooth featureless cloud bank. The rainbow cone intersects the horizontal cloud layer in a hyperbola if ◦ ◦ the sun’s elevation is less than 42 (or an ellipse if it is greater than 42 ) as familiarity with the conic sections assures us. This phenomenon can also be seen as a “dewbow” on a lawn when the sun is low in the eastern sky. Other related phenomena of interest, such as reTected-light rainbows (produced by reTection from a surface of water behind the observer) and reTected rainbows (produced by reTection from a surface of water in front of the observer) can be found in the book by Greenler [4]. Another feature of rainbows produced by smaller drops is also related to the wave nature of light. This time the phenomenon is interference and it produces supernumerary bows. These are a
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series of faint pink or green arcs (2– 4 perhaps) just beneath the top of the primary bow, or much more rarely, just above the top of the secondary bow. They rarely extend around the fully visible arc, for reasons that are again related to drop size. Two rays that enter the drop on either side of the rainbow ray (the ray of minimum deviation) may exit the drop in parallel paths; this certainly will happen for appropriately incident rays. By considering the wavefronts (perpendicular to the rays), it is clear that if the incident waves are in phase (i.e. crests and troughs aligned with crests and troughs) the emerging rays will not be in phase, in general. Inside the drop they travel along paths of di9erent length. Depending on whether this path di9erence is an integral number of wavelengths or an odd integral number of half-wavelengths, these waves will reinforce each other (constructive interference) or annihilate each other (destructive interference). Obviously partially constructive=destructive interference can occur if the path di9erence does not meet the above criteria exactly. Where waves reinforce one another, the intensity of light will be enhanced; conversely, where they annihilate one another the intensity will be reduced. Since these beams of light will exit the raindrop at a smaller angle to the axis than the D7escartes ray, the net e9ect for an observer looking in this general direction will be a series of light and dark bands just inside the primary bow. The angular spacing of these bands depends on the size of the droplets producing them (see Fig. 4). The width of individual bands and the spacing between them decreases as the drops get larger. If drops of many di9erent sizes are present, these supernumerary arcs tend to overlap somewhat and smear out what would have been obvious interference bands for droplets of uniform size. This is why these pale blue or pink or green bands are then most noticeable near the top of the rainbow: it is the smaller drops that contribute to this part of the bow, and these may represent a rather narrow range of sizes. Nearer the horizon a wide range of drop size contributes to the bow, but as we have seen, at the same time it tends to blur the interference bands. There are many complementary levels of mathematical techniques with which one can describe the formation and structure of the rainbow. In this section we examine the broad features using only elementary calculus; one account of the basic mathematics is described in [16] (and a useful graphical account is provided in [17]); a thorough treatment of the related physics of multiple rainbows may be found in the article by Walker [14]. More mathematical details are found in two classical works, one by Humphreys [18] and the other by Tricker [8]. In [16] the description of the location and color of the rainbow is approached as an exercise in mathematical modeling: the laws of reTection and refraction are stated along with the underlying assumptions about the deviation of sunlight by a raindrop (e.g. the sphericity of the drop; parallel light “rays” from the sun, neither of which are strictly true, of course; the raindrop being Axed during the scattering process, amongst other ‘axioms’). At this elementary level of description involving geometrical optics, dispersion, geometry, trigonometry and calculus of a single variable, a reasonably satisfactory “explanation” of the main features of the rainbow is possible. Incorporation of wave interference and di9raction, amongst other things, is necessary to take the description further, and as we shall see, this can lead to very sophisticated models indeed. A most fascinating aspect of the topic at hand is the fact that rainbows and glories can also be produced in atomic, molecular and nuclear scattering experiments, thus illustrating at a profound level the wave=particle duality of matter and radiation. Rays in geometrical optics become
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Fig. 4. A wavefront version of the ray geometry shown in Figs. 2 and 3; based on work by Fraser [141] this shows the folded wavefronts emanating from a spherical raindrop of radius a for di9erent size parameters = 2 a= = ka: Here k is the wavenumber of the light; note that the fringes correspond to the supernumerary bows, and that larger drops (here = 250) produce narrow, closely spaced supernumeraries whereas smaller drops ( = 75) produce broader, more widely spaced ones.
particle trajectories, and the refraction of such rays corresponds to particle deTection under the inTuence of atomic or nuclear forces. This puts a new perspective on Hamilton’s recognition of the analogy between geometrical optics and classical particle mechanics, an analogy that has had a powerful inTuence on the development of mathematical physics since the mid-19th century. There are obvious di9erences between “trajectories” in geometrical optics, which are line segments with discontinuities in direction, and those in particle scattering systems, which are curved and smoothly varying, in fact, di9erentiable with respect to arc length or impact parameter. This latter parameter is the distance of an incident ray or particle trajectory from the central axis of the droplet or scattering object, and in the case of the optical rainbow it lies between zero and the droplet radius. In scattering problems it can extend in principle to inAnity. There is a one-to-one correspondence between impact parameter and deTection angle (see Appendix A), and there is a unique trajectory for which the (local) angular deTection is a maximum.
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By analogy this is the “rainbow angle” for the interaction because a concentration of scattered particles arises near this angle. However, the analogy extends much further than this, for as will be noted in Section 4, Ford and Wheeler [19] carried out a wave–mechanical analysis of atomic and nuclear rainbows, showing that interference between trajectories emerging in the same direction gives rise to supernumerary bows (peaks in intensity). Furthermore, it has been possible to formulate an analogue of Airy’s theory (Section 2) for particle scattering. In 1964, Hundhausen and Pauly [20] made the Arst observations of an atomic rainbow with an experiment in which sodium atoms were scattered by mercury atoms. The “primary” bow (there is no secondary) and two supernumeraries were detected; subsequent experiments have revealed structure at yet Aner scales. Just as careful observations of optical rainbows yield information on the scatterering centers—the raindrops, the refractive index, etc., so too do experiments such as these. The atomic (and for that matter, molecular and nuclear) rainbow angle depends on the strength of the interaction, speciAcally the attractive part, and the range of the interaction in turn determines the positions of the supernumerary bows. This topic will be discussed at greater length in Section 6.5. Returning to the optical rainbow, it is important to note that the theories of D7escartes, Newton and even Young’s interference theory all predicted an abrupt transition between regions of illumination and shadow (as at the edges of Alexander’s dark band when rays only giving rise to the primary and secondary bows are considered). In the wave theory of light such sharp boundaries are softened by di9raction—and this should have been Young’s conclusion, incidentally, for his was a wave theory [2]. In 1835, Potter showed that the rainbow ray can be interpreted as a caustic, i.e. the envelope of the system of rays comprising the rainbow. The word caustic means “burning curve”, and caustics are associated with regions of high intensity illumination (as we shall below, geometrical optics predicts an inAnite intensity there). When the emerging rays comprising the rainbow are extended backward through the drop, another caustic (a virtual one) is formed, associated with the real caustic on the illuminated side of the drop. Typically, the number of rays di9ers by two on each side of a caustic at any given point, so the rainbow problem is essentially that of determining the intensity of (scattered) light in the neighborhood of a caustic (see Fig. 3, and also Fig. 5). This was exactly what Airy attempted to do several years later in 1838 [21]. The principle behind Airy’s approach was established by Huygens in the 17th century: Huygens’ principle regards every point of a wavefront as a secondary source of waves, which in turn deAnes a new wavefront and hence determines the subsequent propagation of the wave. Airy reasoned that if one knew the amplitude distribution of the waves along any complete wavefront in a raindrop, the distribution at any other point could be determined by Huygens’ principle. However, the problem is to And the initial amplitude distribution. Airy chose as his starting point a wavefront surface inside the raindrop, the surface being orthogonal to all the rays which comprise the primary bow; this surface has a point of inTection wherever it intersects the ray of minimum deviation—the rainbow ray. Using the standard assumptions of di9raction theory, he formulated the local intensity of scattered light in terms of a “rainbow integral”, subsequently renamed the Airy integral in his honor (it is related to the now familiar Airy function; see Appendix B. Note that there is another so-called Airy function in optics which has nothing to do with this phenomenon [22]). Qualitatively, the intensity distribution so produced is similar to that associated with the shadow of a straight edge, particularly when external reTection is included (see [23, Chapter 8]).
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Fig. 5. A caustic curve (bold curve CC) showing rays and corresponding wavefronts on the “lit” side; in simplistic terms the caustic is a shadow boundary, but in reality there are transition regions that smoothly blend regions of darkness and light (see Section 5).
One severe limitation of the Airy theory for the optical rainbow is that the amplitude distribution along the initial wavefront is unknown: based on certain assumptions it has to be guessed. There is a natural and fundamental parameter, the size parameter, ; which is useful in determining the domain of validity of the Airy approximation; it is deAned as the ratio of the droplet circumference to the wavelength of light ( ). In terms of the wavenumber k this is 2 a
= = ka ; a being the droplet radius. Typically, for sizes ranging from fog droplets to large raindrops,
ranges from about 100 to several thousand. Airy’s approximation is a good one only for
& 5000: In the light of these remarks it is perhaps surprising that an exact solution does exist for the rainbow problem, as it is commonly called. The more precisely expressed version is the ‘scattering of plane electromagnetic waves by a homogeneous transparent sphere’, which is in turn related to such scattering by an impenetrable sphere. This latter problem has historical signiAcance in connection with the di9raction of radio waves around the earth, amongst other applications. The same type of formulation arises in the scattering of sound waves by a sphere, studied by Lord Rayleigh and others in the 19th century. The solution to this problem is expressible in terms of an inAnite set of partial waves called, not surprisingly, a partial wave expansion. The corresponding result for electromagnetic wave scattering was developed in 1908 by Mie [24]. It is obviously of interest to determine under what conditions such an inAnite set of terms (each of which is a somewhat complicated expression) can be truncated, and what the resulting error may be by so doing. However, in conArmation of the conjecture that there is no such thing as a free lunch, it transpires that the number of terms that must be retained is of the same order of magnitude as the size parameter, i.e. up to several thousand
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for the rainbow problem! The “why is the sky blue?” scattering problem, on the other hand— Rayleigh scattering—requires only one term because the scatterers are molecules which are much smaller than a wavelength, so the simplest truncation—retaining only the Arst term—is perfectly adequate. Although in principle the rainbow problem can be “solved” with enough computer time and resources, numerical solutions by themselves (as Nussenzveig points out) o9er little or no insight into the physics of the phenomenon. Fortunately help was at hand. The now-famous Watson transformation, developed by Watson [25] is a method for transforming the slowly converging partial-wave series into a rapidly convergent expression involving an integral in the complex angular-momentum plane (see Appendix C). But why angular momentum? Although they possess zero rest mass, photons have energy E = hc= and momentum E=c = h= where h is Planck’s constant and c is the speed of light in vacuo. Thus, for a non-zero impact parameter bi , a photon will carry an angular momentum bi h= , (which can in fact assume only certain discrete values). Each of these discrete values can be identiAed with a term in the partial wave series expansion. Furthermore, as the photon undergoes repeated internal reTections, it can be thought of as orbiting the center of the raindrop. Why complex angular momentum? This allows some powerful mathematical techniques to “redistribute” the contributions to the partial wave series into a few points in the complex plane—speciAcally poles (called Regge poles in elementary particle physics) and saddle points. Such a decomposition means that instead of identifying angular momentum with certain discrete real numbers, it is now permitted to move continuously through complex values. However, despite this modiAcation, the poles and saddle points have profound physical interpretations in the rainbow problem; (this is known in other contexts: in many wave propagation problems poles and branch points of Green’s functions in the complex plane can be identiAed with trapped and freely propagating waves, respectively, or in terms of discrete and continuous spectra of certain Sturm–Liouville operators (see e.g. [26 –28] and references therein). A thoughtful and somewhat philosophical response to the question ‘why complex angular momentum?’ appears in a quotation at the very end of this review (Section 8). Mathematical details will be provided later in Section 5, but for now it will suYce to point out that since real saddle points correspond to ordinary rays of light, we might expect complex saddle points to correspond in some sense to complex rays (whatever that might mean). Again, in problems of wave propagation, imaginary components can often be identiAed with the damping of some quantity—usually the amplitude—in space or time. An elementary but perfect example of this can be found in the phenomenon of (critical) total internal reTection of light at a glass=air or water=air boundary: the “refracted” ray is in e9ect a surface wave, propagating along the interface with an amplitude which diminishes exponentially away from that interface. Such waves are sometimes identiAed as “evanescent”, and are common in many other non-quantum contexts [26]. Typically, the intensity is negligible on the order of a wavelength away from the interface; this deAnes a type of “skin depth”. Quantum-mechanical tunneling has a similar mathematical description, and, directly related to the rainbow problem, complex rays can appear on the shadow side of a caustic, where they represent the damped amplitude of di9racted light. In the light of this (no pun intended) it is interesting to note that there is another type of surface wave: this one is associated with the Regge pole contributions to the partial-wave expansion. Such waves are initiated when the impact parameter is the drop diameter, i.e.
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the incident ray is tangential to the sphere. These waves are damped in the direction of increasing arc-length along the surface, i.e. tangentially, (as opposed to radially, in the case of an evanescent wave) because at each point along its circumferential path the wave penetrates the spherical drop at the critical angle for total internal reTection, in a reversal of the normal event. Subsequently it will re-emerge as a surface wave after taking one or more of these shortcuts. In [2] Nussenzveig points out that this “pinwheel” path for light rays was considered by Kepler in 1584 (see [11,29]) as a possible explanation for the rainbow, but he did not pursue it because he could not reproduce the correct rainbow angle. It is important to note that the Watson transformation was originally introduced in connection with the di9raction of radio waves around the earth [25], alluded to above. In 1937, Van der Pol and Bremmer [30,31] applied it to the rainbow problem, but, notwithstanding a claim by Sommerfeld that the problem had now been brought to a beautiful conclusion [32], they were able only to establish that the Airy approximation could be recovered as a limiting case. (Their papers contain some very beautiful and detailed mathematics, however, and are signiAcant contributions nonetheless.) In some profound and highly technical papers Nussenzveig subsequently developed a modiAcation of the transformation [33,5,6] (a valuable summary of his work up to that point may be found in [34]; references to more recent work will be made throughout the review) and was able to bring the problem to the desired conclusion using some extremely sophisticated mathematical techniques without losing sight of the physical implications. In the simplest Cartesian terms, on the illuminated side of the rainbow (in a limiting sense) there are two rays of light emerging in parallel directions: at the rainbow angle they coalesce into the ray of minimum deTection, and on the shadow side, according to geometrical optics, they vanish (this is actually a good deAnition of a caustic curve or surface; refer again to Fig. 5). From the above general discussion of real and complex rays it should not be surprising that, in the complex angular momentum plane, a rainbow in mathematical terms is the collision of two real saddle points. But this is not all: this collision does not result in the mutual annihilation of these saddle points; instead a single complex saddle point is born, corresponding to a complex ray on the shadow side of the caustic curve. This is directly associated with the di9racted light in Alexander’s dark band. 1.3. The rainbow: elementary mathematical considerations As already noted, the primary rainbow is seen in the direction corresponding to the most dense clustering of rays leaving the drop after a single internal reTection. Correspondingly higher-order rainbows exist (in principle) at similar “ray clusters” for k internal reTections, k = 2; 3; 4; : : : . Although only the primary and secondary rainbow are seen in nature, Walker [14] has determined the scattering angles, widths, color sequences and scattered light intensities of the Arst 20 rainbows using geometric optics (he also examined the associated interference patterns, discussed in Sections 2 and 6 below). Referring to Figs. 1(a) and (b), consider the more general angular deviation Dk between the incident and emergent rays after k internal reTections. Since each such reTection causes a deviation of − 2r; (r = r(i) being the angle of refraction, itself a function of i, the angle of incidence) and two refractions always occur, the
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total deviation is Dk (i) = k( − 2r) + 2(i − r) :
(1.1)
Thus, for the primary rainbow (k = 1); D1 (i) = − 4r + 2i; and for the secondary rainbow (k = 2); D2 = 2i − 6r (modulo 2 ). Seeking an extremum of Dk (i), and using Snell’s law of refraction twice (once in di9erentiated form) it is readily shown that for all k; Dk (i) is an extremum when cos i N = cos r k + 1
or after some rearrangement, 2 1=2 N −1 cos i = ; k(k + 2)
(1.2)
(1.3)
where N is the index of refraction for the incident ray (assumed monochromatic for the present). For this value of i, im say, Dk = Dkm ; so that for k = 1 in particular sin im m D1 = + 2im − 4 arcsin : (1.4) N That this extremum is indeed a minimum for k = 1 follows from the fact that Dk (i) =
2(k + 1)(N 2 − 1)tan r N 3 cos2 r
(1.5)
(see [18]) which is positive since N ¿ 1 and 0 ¡ r ¡ =2, except for the special case of normal incidence, so the clustering of deviated rays corresponds to a minimum deTection. Note also that the generalization of Eq. (1.4) above to k internal reTections is sin im m Dk (im ) = k + 2im − 2(k + 1)arcsin : (1.6) N Returning to the primary rainbow for which k = 1, since the minimum angle of deviation depends W in on N , we now incorporate dispersion into the model. For red light of wavelength 6563 A W N ≈ 1:3435. These correspond water [8], N ≈ 1:3318, and for violet light of wavelength 4047 A; ◦ ◦ to D1m ≈ 137:7 (i.e. a rainbow angle—the supplement of D1m , of ≈ 42:3 —see below) and ◦ ◦ D1m ≈ 139:4 (a rainbow angle of ≈ 40:6 ), respectively. This gives a theoretical angular width ◦ ◦ of ≈ 1:7 for the primary rainbow; in reality it is about 0:5 wider than this [2] because this is the angular diameter of the sun for an earth-based observer, and hence the incident rays can deviate from parallellism by this amount (and more to the point, so can the emergent rays as a simple di9erential argument shows). The above-mentioned rainbow angle for a given ◦ Dkm (or Dmin ) is the supplementary angle 180 −Dkm ; this is the angle of elevation relative the ◦ sun-observer line and for the primary rainbow is about 42 . Light deviated at angles larger than this minimum will illuminate the sky inside the rainbow more intensely than outside it, for this very reason. For this reason, the outside “edge” of the primary bow will generally be ◦ more sharply deAned than the inner edge. For the secondary rainbow (k = 2) D2m = Dmin ≈ 231
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245 ◦
and the rainbow angle in this case is D2m − 180 , or approximately 51 for light of an orange color for which we choose n = 43 (sometimes, as in [8] the angle of minimum deviation is ◦ deAned as the complement of Dmin ; which for k = 2 is ≈ 129 ). Each additional reTection of course is accompanied by a loss of light intensity because of transmission out of the drop at that point, so on these grounds alone it would be expected that the tertiary rainbow (k = 3) ◦ would be diYcult to observe. In this case Dmin ≈ 319 , and the light concentration therefore is ◦ at about an angle of 41 from the incident light direction. This means that it appears behind the observer as a ring around the sun. Due to (i) the increased intensity of sunlight in this region, (ii) the fact that the angles of incidence im increase with k (see (1.3)) and result in a reduction of incident intensity per unit area of the surface, and hence also a reduction for the emergent beam, (iii) higher-order rainbows are wider than orders one and two, (iv) the presence of light reTected from the outer surface of the raindrops (direct glare), (v) light emerging with no internal reTections (transmitted glare), and (vi) the reduced intensity from three reTections, it is not surprising that such rainbows (i.e. k ¿ 3) have not been reliably reported in the literature (but see [13]). At this point it is useful to note the alternative valuable and very succinct treatment of the geometrical theory of the rainbow provided by Jackson [35]. Instead of using the angles of incidence i and refraction r, the author works in terms of the impact parameter b (normalized by the drop radius a so the fundamental variable is x = sin i = b=a; see also the appendix in [16]). For the primary rainbow in particular (k = 1) Eq. (1.1) becomes = + 2 arcsin x − 4 arcsin(x=N ) ;
(1.1a)
where the deviation angle D1 has been replaced by ; which is the standard variable used in the study of scattering cross sections (see Appendix A and Section 1.5 below; it is useful to interpret the rainbow problem in terms of scattering theory even at this basic level of description). Noting that d = d i = cos i d = d x for given N , it follows that extrema of (i) can be expressed in terms of extrema of (x); hence (x) = √
2 4 −√ 2 2 1−x N − x2
(1.1b)
and (x) =
2x 4x − : 2 3=2 2 (1 − x ) (N − x2 )3=2
(1.1c)
By requiring (x0 ) = 0 it follows that x0 = (4 − N 2 )=3 from which result (1.3) is recovered for k = 1: Obviously, this result can be generalized to other positive integer values of k: Note also that
(x0 ) =
9(4 − N 2 )1=2 ¿ 0 for 1 ¡ N ¡ 2 ; 2(N 2 − 1)3=2
so the angle of deviation is indeed a minimum as expected. We note two other aspects of the problem discussed in [35] (this article will be mentioned again in connection with Airy’s theory of the rainbow in Section 2.1). First, it is clear that for x ≈ x0 the familiar quadratic “fold” for
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(x) is obtained: ≈ 0 + (x0 )(x − x0 )2 =2 ;
(1.1d)
where 0 = (x0 ): This result will be required in Section 2.1. The second comment concerns dispersion: since = (x; N ) it follows from (1.1a) that 4x 9 9 =−4 [arcsin(x=N )] = √ 9N 9N N N 2 − x2 ◦
and so at the rainbow angle (0 ≈ 138 corresponding to x0 = 9 2 4 − N2 = : 9N N N 2 − 1
(4 − N 2 )=3 ≈ 0:86 for N = 43 ) (1.1e)
This can be used to estimate the angular spread of the rainbow (]) given the variation in N ◦ over the visible part of the spectrum (]N ); as noted above ] is slightly less than 2 for the primary bow. Note also that since d d dN ≈ d dN d
for given x0 (i.e. neglecting the small variation of x0 with wavelength ) and d N= d ¡ 0; it follows that d = d ¡ 0, and so the red part of the (primary) rainbow emerges at a smaller angle than the violet part, so the latter appears on the underside of the arc with the red outermost. Thus far we have examined only the variation in the deviation of the incident ray as a function of the angle of incidence (or as a function of the normalized impact parameter). Even within the limitations of geometrical optics, there are two other factors to consider. The Arst of these is the behavior of the coeYcients of reTection and refraction as a function of angle of incidence, and the second is to determine how much light energy, as a function of angle of incidence, is deviated into a given solid angle after interacting with the raindrop. This might be thought of as a classical “scattering cross section” type of problem. The Arst of these factors involves the Fresnel equations, and though we shall not derive them here an excellent account of this can be found in the book by Born and Wolf [23] (see also [22,36]: another classic text). 1.4. Polarization of the rainbow Electromagnetic radiation—speciAcally light—is propagated as a transverse wave, and the orientation of this oscillation (or “ray”) can be expressed as a linear combination of two “basis vectors” for the “space”, namely, mutually perpendicular components of two independent linear polarization states [2]. Sunlight is unpolarized (or randomly polarized), being an incoherent mixture of both states, but reTection can and does alter the state of polarization of an incident ray of light. For convenience we can consider the two polarization states of the light incident upon the back surface of the drop (i.e. from within) as being, respectively, parallel to and perpendicular to the plane containing the incident and reTected rays. Above a critical angle of incidence (determined by the refractive index) both components are totally reTected, although some of the light does travel around the surface as an “evanescent” wave; this shall be discussed
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247
Fig. 6. The reTectivity of light impinging on a water=air interface from within the denser medium (i.e. from within the raindrop) as a function of the angle of incidence, resulting in the polarization of the rainbow. There is always some reTection of the component perpendicular to the plane of reTection (the plane containing the incident and reTected rays); the parallel component however, is completely transmitted at the Brewster angle. At the critical angle for total internal reTection, surface waves can be created, travelling along the surface of the drop for several degrees of arc (see Sections 3 and 5). (Redrawn from [2].)
in more detail in a later section. This critical angle of incidence, ic ; from within the drop is deAned by the expression ic = arcsin(N −1 ) ;
(1.7)
where N is the refractive index of the water droplet (more accurately, N is the ratio of the refractive indices of water and air, both relative to a vacuum). Taking a generic value for N of 43 ◦ (orange light) we And ic ≈ 48:6 : At angles i less than ic the parallel component of polarization is reTected less eYciently than its perpendicular counterpart, and at the Brewster angle it is entirely transmitted, leaving only perpendicularly polarized light to be reTected (partially) from the inside surface of the drop. Since the Brewster angle, as shown below, is close to the angle of total internal reTection ic , the light which goes on to produce the rainbow is strongly perpendicularly polarized. At the Brewster angle the reTected and refracted rays are orthogonal (and hence complementary), and therefore it follows from Snell’s law that iB = arctan(N −1 )
(1.8) ◦
◦
(remember this is from within the drop), from which iB ≈ 36:9 ; di9ering from ic by about 12 (see Fig. 6 and also [2]). It is perhaps worth pointing out here that the discussion by Tricker [8] is for an air-to-water interface, so that N −1 in Eq. (1.8) above should be replaced by N ; ◦ the corresponding value iB above is the supplement of the above angle, i.e. ≈ 53:1 :
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The coeYcient of reTection of light depends on its degree of polarization: Consider Arst the case of light polarized perpendicular to the plane of incidence, with the amplitude of the incident light taken as unity. It follows from the Fresnel equations that the fraction of the incident intensity reTected is [23] sin2 (r − i) sin2 (r + i)
(1.9)
and tan2 (r − i) ; tan2 (r + i)
(1.10)
if the light is polarized parallel to the plane of incidence (there is an apparent discrepancy between the terminology of Tricker [8] and Walker [14] at this point: in particular the diagrams in [8] for the Fresnel coeYcients as a function of i for an air-to-water interface have the opposite sense of polarization to that in [22,23] and other texts). It follows that for a ray entering a drop and undergoing k internal reTections (plus 2 refractions), the fraction of the original intensity remaining in the emergent ray is, for perpendicular polarization, 2 2 2k 2 sin (r − i) sin (r − i) Ik1 = 1 − (1.11) sin2 (r + i) sin2 (r + i) and for parallel polarization 2 2 2k 2 tan (r − i) tan (r − i) : Ik2 = 1 − tan2 (r + i) tan2 (r + i)
(1.12)
The total fraction is the sum of these two intensities. For angles of incidence close to normal, for which i ≈ Nr; it follows that the single-reTection intensity coeYcients for reTection and refraction become, from Eq. (1.9) N −1 2 (1.13) N +1 and 4N ; (N + 1)2
(1.14)
respectively. For the choice of N = 43 under these circumstances it follows that the reTection and refraction coeYcients are approximately 0:02 and 0:98, respectively. An alternative but entirely equivalent formulation of the Fresnel equations is applied to the polarization of the (primary) rainbow in [35], using the relative amplitude equations for polarization both perpendicular to (E⊥ ) and parallel to the plane of incidence (E|| ) [36]. For each polarization, the emerging relative amplitude (i.e. scattered : incident) is determined by the product of the relative amplitudes corresponding to the three interfaces, namely transmission at an air=water interface, reTection at a water=air interface and transmission at a water=air interface (for rays 1, 2, 3 and 3 , respectively, in Fig. 1(c)). Care must be taken to identify the correct refractive indices and angles of incidence, both of which depend on which medium the ray is
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about to enter. The calculations (omitted in [35]) are straightforward but nonetheless worthy of note, and are summarized below. There are three interfaces to consider for the primary rainbow analysis: air=water transmission (with transmission coeYcients ts(1) and tp(1) for the vector E perpendicular and parallel to the plane of incidence, respectively); water=water re9ection (with corresponding reTection coeYcients rs(2) ; rp(2) ), and water=air transmission (with transmission coeYcients ts(3) ; tp(3) ). Each will be considered in turn; in what follows n denotes the refractive index of the ‘incidence’ medium, n that of the ‘transmission’ medium, and as usual, i and r represent the angles of incidence and refraction. N is the refractive index of water; that of air will be taken as unity. The formulae for the various coeYcients are taken from [36]. 1.4.1. Air=water transmission n = 1; n = N = 43 ; at the (primary) rainbow angle (according to geometrical optics) N2 − 1 cos i = 3 from which the remaining quantities follow, so that 2 cos i 2 ts(1) = = 2 cos i + N 2 − sin i 3 and
tp(1) =
N 2 cos i
2N cos i 2N : = 2 2 + N2 2 + N − sin i
1.4.2. Water=water re9ection n = N = 43 ; n = 1; in this case 1 4 − N2 sin r = ; N 3 so that N cos r − 1 − N 2 sin2 r 1 (2) rs = = N cos r + 1 − N 2 sin2 r 3 and cos r − N 1 − N 2 sin2 r 2 − N 2 (2) rp = : = 2 cos r + N 1 − N 2 sin2 r 2 + N 1.4.3. Water=air transmission Again, n = N = 43 ; n = 1; and in this Anal situation 2N cos r 4 ts(3) = = N cos i + 1 − N 2 sin2 r 3 and 2N cos r 4N : = tp(3) = 2 cos i + N 1 − N 2 sin2 r 2 + N
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These coeYcients are combined according to respective products (i) ts(1) rs(2) ts(3) = 23 and tp(1) rp(2) tp(3) =
1 4 3 3
8 = 27
2 2 − N2 2N 2N 2 − N 2 4N =2 2 + N2 2 + N2 2 + N2 2 + N2 2 + N2
so that the required relative amplitudes are Escatt 8 (1) = 27 (for E⊥ ) and (1:15) Einc 2 Escatt 2N 2 − N2 =2 (for E|| ) : Einc N2 + 2 2 + N2 It follows that the relative intensity of the perpendicular polarization (for N = 43 ) is 8:78% (i.e. I⊥ ≈ 0:0878I0 ) and for the parallel polarization it is 0:34% (i.e. I|| ≈ 0:0034I0 ; note therefore, that I|| ≈ 0:039I⊥ ; so the primary bow is about 96% perpendicularly polarized). Notice from Eq. (1.10) that the fraction of the intensity of reTected light is zero (for parallel polarization) when the denominator in the coeYcient vanishes, i.e. when i+r = =2; at this point the energy is entirely transmitted, so the internally reTected ray is completely perpendicularly polarized. From this it follows that sin i = N sin r = N cos i ;
whence tan i = tan iB = N
for external reTection, and Eq. (1.8) for internal reTection. In the latter case, for N = 43 , this ◦ yields iB ≈ 36:9 ; the Brewster polarizing angle, as pointed out above. Note that as i + r “passes through” =2; the tangent changes sign (obviously being undeAned at =2); this corresponds ◦ to a phase change of 180 which will be noted when the so-called “glory” is discussed later. Notwithstanding the di9erences in terminology, it is also shown in [8], using Eqs. (1.11) and (1.12) that the primary rainbow is about 96% polarized (as shown above using Eqs. (1:15)), and the secondary bow somewhat less so: approximately 90%. 1.5. The physical basis for the divergence problem In this subsection the physics of the di;erential scattering cross section is discussed (see the deAnition of this in Appendix A). Referring to Fig. 7, note that for light incident upon the raindrop in the incidence interval (i; i + i) the area “seen” by the incident rays is (2 a sin i)(a cos i) i or a2 i sin 2i :
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Fig. 7. The geometry leading to the derivation of the geometrical optics intensity function (1.15), which predicts inAnite intensity at the rainbow angle. (Redrawn from [8].)
These rays are then scattered into an interval (; + ) say, measured from the direction of incidence, and this results in solid angle of (2 sin ) being occupied by these rays. Thus the energy (relative to incoming energy) entering unit solid angle per unit time after internal reTection (and neglecting losses from refraction and reTection) is a2 sin 2i i : (1.15) 2 sin Passing to the limit, the Cartesian condition for the occurrence of the rainbow is of course that
d = d i = 0, but now we are in possession of a geometric factor which modulates non-zero values
of this derivative. This factor is undeAned when = 0 or (these cases occurring when i = 0), but for the case of a single internal reTection with both i and “small”, using = − D = 4r − 2i, we have lim
i→0+
sin 2i 2i 2i N = = = : sin 4r − 2i 2 − N
(1.16)
Since this ratio is just 2 when N = 43 it is clear that the intensity of the “backscattered” light is not particularly large when = 0. This excludes, of course, any di9raction e9ects. Note that ◦ for large angles of incidence (i.e. i ≈ 90 ), the angle (as deAned in Fig. 7) never returns to zero, being given instead by the expression ◦
= 4 arcsin(N −1 ) − 180 ;
(1.17)
which for N = 43 is approximately 14:4 according to geometrical optics (more will be said about such grazing incidence rays when the glory is discussed in Section 3). Of course, it has already ◦ been noted that at the rainbow angle (corresponding to ≈ 42 ) the intensity is predicted to become inAnite according to the same theory! ◦
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2. Theoretical foundations 2.1. The supernumerary rainbows; a heuristic account of Airy theory These bows are essentially the result of interference between rays emerging from the raindrop close to the rainbow angle (i.e. angle of minimum deviation). In general they will have entered at di9erent angles of incidence and traversed di9erent paths in the denser medium; there is of course a reduction in wavelength inside the drop, but the overall e9ect of di9erent path lengths is the usual “di9raction” pattern arising as a result of the destructive=constructive interference between the rays. The spacing between the maxima (or minima) depends on the wavelength of the light and the diameter of the drop—the smaller the drop the greater the spacing (see Fig. 4). Indeed, if the drops are less than about 0:2 mm in diameter, the Arst maximum will be distinctly below the primary bow, and several other such maxima may be distinguished if conditions are conducive. Although this phenomenon is decidedly a “wavelike” one, we can gain some heuristic insight into this mechanism by considering, following Tricker [8], a geometrical optics approach to the relevant rays and their associated wavefronts. This is done by considering, not the angle of minimum deviation, but the point of emergence of a ray from the raindrop as the angle of incidence is increased. A careful examination of Fig. 1(a) reveals that as the angle of incidence is increased away from zero, while the point of entry moves clockwise around the drop, the point of emergence moves Arst in a counterclockwise direction, reaches an extreme position, and then moves back in a clockwise direction. This extreme point has signiAcant implications for the shape of the wavefront as rays exit the drop in the neighborhood of this point. Referring to this Agure, we wish to And when angle BOA = 4r − i is a maximum. This occurs when dr 1 = di 4 and d2 r ¡0 : d i2
The Arst derivative condition leads to
or
cos2 i = 16 N 2 − sin2 i
(2.1)
1=2 2 N −1 i = arccos ;
15
(2.2)
whence for N = 43 , i ≈ 76:8 , well past the angle of incidence corresponding to the rainbow ◦ angle, which is i ≈ 59:4 for the same value of N . As can be seen from Fig. 8(a), if YEY represents the ray which emerges at minimum deviation, rays to either side of this are deviated through larger angles as shown. By considering the corresponding wavefront WEW (distorted from the wavefront XOX (see Fig. 8(b)) corresponding to a parallel beam of rays) it is clear that we are dealing with, locally at least, a cubic approximation to the wavefront in the neighborhood ◦
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Fig. 8. (a) The ray of minimum deviation YY and nearby rays emerging from the (locally) cubic wavefront WW , (b) the geometry of the wavefront WW leading to the Airy integral form for the amplitude (2.8). (Redrawn from [8].)
of the point E, which is itself a point of inTection. With point E as the origin of coordinates and YY , XX as the coordinate axes, it is clear that, as drawn, the equation of the wavefront WEW can be written as y = c x 3 ; since y (0) = 0, where c is a constant with dimensions of (length)−2 . It is reasonable to expect that the linear dimensions of the wavefront will be related to the size of the raindrop from whence it comes, so in terms of the drop radius a we write c y = 2 x3 ; (2.3) a c now being dimensionless. At this point, we are in a position to deduce the form of the famous “rainbow integral” introduced by Sir George Airy in 1838 [21]. Using Fig. 8(b) we can derive an expression for the amplitude of the wave in a direction making an angle with that of minimum deviation. By considering the path di9erence % between the points P(x; y) and O we have % = OS = OR − RS = OR − TN = x sin − y cos or cx3 cos : (2.4) a2 In relative terms, if the amplitude of a small element x of the wavefront at O is represented by sin $t, then that from a similar element at P is sin($t + ), where 2 % = = k% ; % = x sin −
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being the wavelength, and k being the wavenumber of the disturbance. For the whole wavefront the cumulative disturbance amplitude is therefore the integral
∞ A= sin($t + k%) d x −∞
or, since sin k% is an odd function,
∞ A = sin $t cos k% d x : −∞
(2.5)
In terms of the following changes of variable: +3 =
2kcx3 cos a2
(2.6)
2kx sin ;
(2.7)
and m+ =
the above integral may be written in the canonical form 1=3 ∞ a2 cos (m+ − +3 ) d + : 2kc cos 2 −∞
(2.8)
This is Airy’s rainbow integral, Arst published in his paper entitled “On the Intensity of Light in the neighbourhood of a Caustic” [21]. The signiAcance of the parameter m can be noted by eliminating + from Eqs. (2.6) and (2.7) to obtain 1=3 2ka 2=3 sin3 m= ; (2.9) cos which for suYciently small values of (the angle of deviation from the rainbow ray) is proportional to 3 . A graph of the rainbow integral qualitatively resembles the di9raction pattern near the edge of the shadow of a straight edge, which has the following features: (i) low-intensity, rapidly decreasing illumination in regions that geometrical optics predicts should be totally in shadow, and (ii) in the illuminated region (as with di9raction), a series of fringes, which correspond to the supernumerary bows below the primary rainbow. The Arst maximum is the largest in amplitude, and corresponds to the primary bow; the remaining maxima decrease rather slowly in amplitude, the period of oscillation decreasing also. The underlying assumption in this approach is that di9raction arises from points on the wavefront in the neighborhood of the D7escartes ray (of minimum deviation); provided that the drop size is large compared to the wavelength of light this is reasonable, and is in fact valid for most rainbows. It would not be as useful an assumption for cloud or fog droplets which are considerably smaller than raindrops, but even then the drop diameter may be Ave or ten times the wavelength, so the Airy theory is still useful. Clearly, however, it has a limited domain of validity.
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Fig. 9. (a) Details of the cusped wavefront upon its emergence from the raindrop. (b) Interference between the two arms of the cusped wavefronts (greatly exaggerated) can be expected to produce the supernumerary bows. (see Fig. 4 and [8], from which these Agures are redrawn.)
Another feature of the rainbow problem that has been neglected thus far is the threedimensional nature of the di9raction: the wavefront is a surface in three dimensions, not merely a curve in two. The factor in Eq. (2.8) above, namely 1=3 1=3 a2 a2 = ; (2.10) 2kc cos 4c cos requires modiAcation. In fact, it is necessary to multiply this factor by (a= )1=2 (see Appendix I in [8] for a discussion of this factor based on Fresnel zones). Since the angle will be small, this results in an amplitude proportional to (a7 = )1=6 or equivalently, an intensity proportional to (a7 = )1=3 . Thus, the relatively strong dependence of intensity upon drop size is established within the Airy regime: other things being equal, larger drops give rise to more intense rainbows. Note also that the -dependence is important to determine the intensity distribution with wavelength. The di9raction pattern may be thought of as arising from interference of two arms of a cusped wavefront [8] (see Figs. 9(a) and (b); but note also the discussion in [37]). A set of maxima and minima occur, lying between the direction of the incident light and the D7escartes ray. A change of phase occurs when light passes a focus (see Fig. 9(a)), so it is to be expected that the D7escartes ray would not correspond exactly to the direction of maximum intensity (which is displaced inwards), and this is indeed the case, as more complete theory shows. Furthermore, it is not the case that the intensities should be the same along both arms of the cusped wavefront, which implies that the minima will not in general be of zero intensity; in Airy’s theory there is assumed to be no variation in intensity along the wave, and as a result the intensity minima are zero. A more mathematical description has been provided in [35] which is both brief and valuable. Referring to Fig. 1(a), we write the phase , accumulated along the critical ray between the surfaces A A and B B is, in terms of the relative impact parameter (x = sin i = b=a) introduced in
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Section 1.3, ,(x) = ka{2(1 − cos i) + 4N cos r } = 2ka{1 − 1 − x2 + 2 N 2 − x2 } : The Arst term inside the Arst set of brackets is the sum of the distances from both A and B to the surface of the drop; the second term is N times the path length interior to the drop. Noting that x 2x , x(x) = 2ka √ −√ 1 − x2 N 2 − x2 and comparing this with the result derived in Section 1 for the derivative with respect to x of the deTection angle (x) it is seen that , (x) = kax (x) : Near the critical angle = 0 (and hence near x = x0 ) this result may be written in terms of the variable + = x − x0 , i.e. , (+) = ka[x0 (+) + + (+)] ; whence
,(+) = ,0 + ka x0 ( − 0 ) + +(+) −
0
+
(+ ) d +
:
Using the result, previously derived, that near = 0 , (+) ≈ 0 + (0 )+2 =2 + O(+3 ) ; it is readily shown that ,(+) ≈ ,0 + ka[x0 ( − 0 ) + (0 )+3 =3 + O(+4 ) ; so that for two rays, each one close to, but on opposite sides of the critical ray, with equal and opposite +-values, it follows that their phase di9erence is = ,(+) − ,(−+) ≈ 2ka (0 )+3 =3 : When a phase di9erence is equal to an integer multiple of 2 then the rays interfere constructively in general; however, in this instance (and as noted above) an additional =2 must be added because a focal line is passed in the process (see [7, chapters 12 and 13]). Hence for constructive interference 2=3
3 (K + 14 ) (0 ) 1=3 K − 0 ≈ ; K = 1; 2; : : : : 8 ka Clearly, within this approximation, K − 0 ˙ (ka)−2=3 which is quite sensitive to droplet size. If the droplets are large enough, the supernumerary bows lie inside the primary bow and thus
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are not visible; Jackson demonstrates that the maximum uniform droplet size rendering them visible is a 0:28 mm [35]. If the droplets are not uniform in size, the maxima may be washed out by virtue of the spread in sizes; if the droplets are all very small (a ¡ 50 m) the various colors are dispersed little and “whitebows” or “cloudbows” result. To capitalize on the expressions for the phase function ,(x) and − 0 , refer now in Agure (∗ ) to the wavefront along BB ; following [35], it can be shown using the Kircho9 integral for di9raction that the amplitude of the scattered wave near = 0 is
∞ 3 ˙ eika[(−0 )+− + =6] d + : scatt −∞
This can be expressed in the form of an Airy integral Ai(−%) (essentially equivalent to Eq. (2.8), which is the form given in [7]), where
1 3 1 ∞ Ai(−%) = cos + − %+ d + ; 0 3 where
%=
2k 2 a2 (0 )
1=3
( − 0 ) :
For more details of the Airy function and the relationship between the two forms, see Appendix B. For large values of % ¿ 0 the dominant term in the asymptotic expansion for Ai(−%) is 2 3=2 Ai(−%) ( 2 %)−1=4 sin % + 3 4 corresponding to slowly decaying oscillations on the “bright” side of the primary bow (for % ¡ 0, Ai(−%) decreases to zero faster than exponentially; this is the “shadow” side of the primary bow). Noting that [35] √ |Ai(−%)|2 = (2 %)−1 ; where : means the average value of its argument, it may be veriAed that 2 1=3 1 ( ) 2 0 |Ai(−%)|2 = : 2 4ka (0 )( − 0 ) Near = 0 , the classical di9erential scattering cross section has been found to be 2 d/ a x0 2 ≈ d 0 class sin 0 | (0 )( − 0 )| (the factor a2 appearing because x is dimensionless). By comparing this directly with the mean square Airy intensity, an approximate expression for the “Airy di9erential cross section” can be inferred, namely d/ 2 a2 x0 4ka 1=3 ≈ |Ai(−%)|2 : d 0 Airy sin 0 2 (0 )
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If the intensity loss from two refractions and one reTection are ignored (they cannot be) this cross section is about 30 times as great as that for an isotropic di9erential cross section (d /= d 0 = a2 =4) at the peak of the primary bow when ka = 103 . This must be multiplied by the scattered=incident ratios calculated in Section 1.4 when these intensity losses are included. This renders the two cross sections comparable in magnitude [35]. The extension of the Airy theory to incorporate the superposition of light of all colors was Arst carried out by Pernter and Exner [38] in 1910, and later by Buchwald [39], who corrected some errors in the former work. The methods and results for the Airy–Pernter theory are well summarized by Tricker, and only a few speciAc results will be noted here. From the basic di9raction theory discussed so far, it is reasonable to expect supernumerary bows to exist outside the secondary bow (in the region of low-intensity illumination). Because of the width of the secondary—more than twice that of the primary—the displacements of the individual color interference patterns will be larger, and whiter, weaker supernumerary bows are to be expected, which, coupled with the lower intensity of the secondary bow means that such supernumeraries are rarely observed. Another point to be mentioned here involves polarization: changes in polarization can be neglected only if the plane of incidence of sunlight on the drop varies by a small angle, which in turn means that integral (2.8) should extend (in practice) only over those portions of the wavefront in the neighborhood of the D7escartes ray. This is clearly another limitation of the theory as originally formulated. Malkus et al. attempted to improve upon the Airy–Pernter theory by developing a better approximation to the shape of the wavefront (amongst other things), subsequently applying their theory to cloud (as opposed to rain) droplets, with mixed success [40]. 2.2. Mie scattering theory Mie theory is based on the solution of Maxwell’s equations of electromagnetic theory for a monochromatic plane wave from inAnity incident upon a homogeneous isotropic sphere of radius a. The surrounding medium is transparent (as the sphere may be), homogeneous and isotropic. The incident wave induces forced oscillations of both free and bound charges in synchrony with the applied Aeld, and this induces a secondary electric and magnetic Aeld, each of which has a components inside and outside the sphere [41] (see also [9,10]). In what follows reference will be made to the intensity functions i1 ; i2 , the Mie coeYcients an ; bn and the angular functions n ; 1n . The former pair are proportional to the (magnitude)2 of two incoherent, plane-polarized components scattered by a single particle; they are related to the scattering amplitudes S1 and S2 in the notation of Nussenzveig [10]. The function i1 ( ; N; ) is associated with the electric oscillations perpendicular to the plane of scattering (sometimes called horizontally polarized) and i2 ( ; N; ) is associated with the electric oscillations parallel to the plane of scattering (or vertically polarized). The scattered wave is composed of many partial waves, the amplitudes of which depend on an ( ; N ) and bn ( ; N ). Physically, these may be interpreted as the nth electrical and magnetic partial waves, respectively. The intensity functions i1 ; i2 are represented in Mie theory as a spherical wave composed of two sets of partial waves: electric (an ) and magnetic (bn ). The Arst set is that part of the solution for which the radial component of the magnetic vector in the incident wave is zero; in the second
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set the corresponding radial component of the electric vector is zero. A given partial wave can be thought of as coming from an electric or a magnetic multipole Aeld, the Arst wave coming from a dipole Aeld, the second from a quadrupole, and so on [41]. The angular functions n () and 1n () are, as their name implies, independent of size ( ) and refractive index (N ). These are investigated in some detail by Penndorf [41] in connection with scattering in the forward, backward and side directions for N = 1:33. For a point P located a distance r from the origin of coordinates, at polar angle and azimuthal angle , the scattered intensities I and I, are, respectively, 2 i2 cos2 , (2.11) I = kr and
I, =
i1 kr
2
sin2 , ;
(2.12)
where ij = |Sj |2 ; j = 1; 2 and the amplitude functions Sj are given by S1 =
∞ 2n + 1 n=1
n(n + 1)
[an n (cos ) + bn 1n (cos )]
(2.13)
[an 1n (cos ) + bn n (cos )] ;
(2.14)
and S2 =
∞ 2n + 1 n=1
n(n + 1)
where n is the order of the induced electric or magnetic multipole. The Legendre functions n (cos ) and 1n (cos ) are deAned in terms of the associated Legendre functions of the Arst kind, Pn1 (cos ) as n (cos ) =
Pn1 (cos ) sin
(2.15)
1n (cos ) =
d 1 P (cos ) : d n
(2.16)
and
The scattering coeYcients an and bn are deAned in terms of the so-called Ricatti–Bessel functions of the Arst and second kinds, respectively, these being z z n (z) : Jn+1=2 (z) and 4n (z) = (−1) J n (z) = 2 2 −(n+1=2) Also note that a Riccati–Hankel function is readily deAned by +n (z) =
n (z)
+ i4n (z) :
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Then it follows that an and bn can be written an =
n ( ) n (5) +n ( ) n (5)
−N −N
n (5) n ( ) n (5)+n ( )
bn =
n (5) n ( ) +n ( ) n (5)
−N −N
n ( ) n (5) n (5)+n ( )
(2.17)
and ;
(2.18)
where the usual dimensionless size parameters are = ka and 5 = N . This form can be simpliAed by the introduction of phase angles; and this results in considerable simpliAcation if the refractive index is real. Further details of the analysis may be found in the book by van de Hulst [7]; for an alternative but entirely equivalent formulation see [10]. Van de Hulst demonstrated that the Mie formulae lead, for large values of ; to a principle for localizing rays and separating di9racted, refracted and reTected light (in the sense of geometrical optics). This is called the localization principle and will be utilized in connection with the theories of the glory in Section 3 and the CAM approach to rainbows and glories in Section 5. The principle asserts that the term of order n in the partial wave expansion corresponds approximately to a ray of distance (n + 12 )=k from the center of the particle (this is just the impact parameter). When
1, the expansions for the Sj may be truncated at n + 12 (in practice, nmax ∼ + 4 1=3 + 2; see [5,10,42]) and the remaining sum is separated into two parts: a di9racted light Aeld component independent of the nature of the particle, and reTected and refracted rays dependent on the particle (see also [43– 46]). In the former case both amplitude functions are equal to the quantity
2
J1 ( sin ) ; ( sin )
(2.19)
a result we shall have occasion to refer to later. 3. Glories 3.1. The backward glory Mountaineers and hill-climbers have noticed on occasion that when they stand with their backs to the low-lying sun and look into a thick mist below them, they may sometimes see a set of colored circular rings (or arcs thereof) surrounding the shadow of their heads. Although an individual may see the shadow of a companion, the observer will see the rings only around his or her head. This is the phenomenon of the glory, sometimes known at the anticorona, the brocken bow, or even the specter of the Brocken (it being frequently observed on this high peak in the Harz mountains of central Germany). A brief but very useful historical account (based on [38]), along with a summary of the theories, can be found in [47]. Early one morning in 1735, a small group of people were gathered on top of a mountain in the Peruvian Andes, members of a French scientiAc expedition, sent out to measure a degree of longitude, led by Bouguer
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and La Condamine; a Spanish captain named Antonio de Ulloa also accompanied them. They saw an amazing sight that morning. According to Bouguer [38] this was “a phenomenon which must be as old as the world, but which no one seems to have observed so far: : : A cloud that covered us dissolved itself and let through the rays of the rising sun: : : Then each of us saw his shadow projected upon the cloud: : : The closeness of the shadow allowed all its parts to be distinguished: arms, legs, the head. What seemed most remarkable to us was the appearance of a halo or glory around the head, consisting of three or four small concentric circles, very brightly colored, each of them with the same colors as the primary rainbow, with red outermost.” Ulloa gave a similar description and also drew a picture. In his account he said “The most surprising thing was that, of the six or seven people that were present, each one saw the phenomenon only around the shadow of his own head, and saw nothing around other people’s heads.” During the 19th century, many such observations of the glory were made from the top of the Brocken mountain in central Germany, and it became known as the “Specter of the Brocken”. It also became a favorite image among the Romantic writers; it was celebrated by Coleridge in his poem “Constancy to an Ideal Object”. Other sightings were made from balloons, the glory appearing around the balloon’s shadow on the clouds. Nowadays, while not noted as frequently as the rainbow, it may be seen most commonly from the air, with the glory surrounding the shadow of the airplane. Once an observer has seen the glory, if looked for, it is readily found on many subsequent Tights (provided one is on the shadow side of the aircraft!). Some beautiful color photographs have appeared in the scientiAc literature [4,48,49]. The phenomenon in simplest terms is essentially the result of light backscattered by cloud droplets, the light undergoing some unusual transformations en route to the observer, transformations not predictable by standard geometrical optics (see the summary of contributing factors listed below). In a classic paper ([50]; later incorporated into his book [7]) van de Hulst put forward his theory of the glory, though at that time the term anti-corona was commonly used for this phenomenon. Noting that glories typically occur for cloud drop diameters of the order of 25 m ( ≈ 150), he used the asymptotic forms for the Legendre functions (spherical harmonics) n and 1n above, namely n (cos ) = (−1)n−1 12 n(n + 1){J0 (z) + J2 (z)}
(3.1)
1n (cos ) = (−1)n 12 n(n + 1){J0 (z) − J2 (z)} ;
(3.2)
and
where as before, n is the order of the induced electric or magnetic multipole, z = (n + 12 )6 and 6 = − is small (note that in [42] z is deAned as n6). Under these circumstances the amplitude functions S1 () and S2 () in the backward direction have the form ◦
◦
S1 (180 ) = − S2 (180 ) =
∞ 1 n=1
2
(2n + 1)(−1)n (bn − an ) :
(3.3)
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Fig. 10. An annular wavefront ABCDE emerging from a water droplet, giving rise to di9racted light at P (in particular). SO is the direction of the incident light, orthogonal to AOB. The geometry of this Agure is used to determine the intensity of the glory in a given direction (see Eq. (3.8)). (Redrawn from [8].)
Debye (see details in [51]) noted that terms in series (2.13) and (2.14) are comparable up to n = and decrease rapidly thereafter, and for the moderately large values of of interest here, most terms in the above sums will nearly cancel out each other; however, this will not be the case when near some value N˜ say, terms of the order : : : N˜ − 3; N˜ − 2; N˜ − 1; N˜ ; N˜ + 1; N˜ + 2; N˜ + 3; : : : have nearly equal phases, resulting in a signiAcant total amplitude. This is essentially a principle of “stationary phase” for sums, as opposed to integrals. Van de Hulst deAnes c1 = (2n + 1)(−1)n bn and c2 = (2n + 1)(−1)n−1 an ; (3.4) n
n
where the sums are to be taken only over those terms in the immediate “neighborhood” of n = N˜ , further terms being neglected. For small 6 values the total amplitude functions can be written subsequently as ◦
S1 (180 − 6) = 12 c2 {J0 (u) + J2 (u)} + 12 c1 {J0 (u) − J2 (u)}
(3.5) ◦
with the c1 and c2 being interchanged for the amplitude S2 (180 − 6), where u = N6. The basic calculations for this phenomenon are based on the intensity of di9racted light arising from an annular (in fact toroidal) wavefront, the radius of which is of the same order of magnitude as the size of the drop [8]. By considering incident light as depicted in Fig. 10, and the disturbance arising from an element of the wavefront at point C in the Agure, it can be shown that [8] the amplitude of the component polarized parallel to OE from this element is a1 = c1 sin(, − 5)sin , + c2 cos(, − 5)cos , ; where c1 and c2 are the amplitudes of the emergent polarizations of the parallel (to plane of incidence) and transverse electric Aelds, respectively. The path di9erence between points C and
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E in the direction OP is = r sin , sin , and the phase di9erence is 2 = ≡ x sin ,. The total amplitude polarized parallel to OE from the whole wavefront is, after some reduction (and using antisymmetry of the cosine between the Arst and second quadrants)
2
2 1 − cos 2, 1 + cos 2, A1 = c1 cos 5 cos(x sin ,) d , + c2 cos 5 cos(x sin ,) d , ; 2 2 0 0 which to within a multiplicative factor is A1 = cos 5(c1 {J0 (x) − J2 (x)} + c2 {J0 (x) + J2 (x)}) :
(3.6)
Similarly, for the perpendicular polarization A2 = sin 5(c1 {J0 (x) + J2 (x)} + c2 {J0 (x) − J2 (x)}) :
(3.7)
For unpolarized light the direction of polarization changes rapidly and randomly, so little is lost by replacing the terms cos2 5 and sin2 5 in the intensity by their averages ( 12 ) to obtain I = A21 + A22 = (c1 + c2 )2 J02 (x) + (c1 − c2 )2 J22 (x) :
(3.8)
The observational consequences of various choices of c1 and c2 (e.g. c1 = ± c2 ; c1 = 0) can be found in Section 13:33 of [7]. Tricker [8] makes the choice c2 = − 0:204c1 based on a rainbow with 96% polarization in the plane of incidence (c22 =c12 = 4=96), and incorporating the change in phase as the Brewster angle is exceeded (see also [50]). Fig. 10 is based on his calculations. Further descriptive (and also historical) details may be found in [52]. More can be said about the glory from a purely geometrical optics viewpoint. Recall that, for the heuristic description of the rainbow (Section 1.5) an important geometrical factor in the amplitude was shown to be sin 2i d i 1=2 : sin d As noted already, rainbows correspond (at this level of description) to the condition d = d i = 0, i.e. inAnite intensity is predicted at the rainbow angle when interference is neglected. Such a divergence also occurs when sin = 0 while sin 2i = 0; which occurs for = 0 and . The features associated with the forward direction are blended with the stronger coronae caused by the di9raction of light around the spherical drop [7], but the backward direction corresponds to the glory. Notice from Fig. 11 that the plane wavefront of the incident wave is emitted as a circular front, which has a virtual focus at F; when the Agure is rotated about the axis of symmetry, the emitted wavefront is seen to be toroidal. Furthermore, this wavefront apparently emerges from the focal circle deAned by Huygens’ principle can be used to describe √the interference pattern corresponding to this wavefront. Note that for a refractive index N ¡ 2 there can be no such o9-axis rays with less than four internal reTections contributing to the glory [7], but the attenuation for these rays is such that their contribution is negligible [53]. Thus according to ray theory, only the axial rays can contribute to this phenomenon.
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Fig. 11. Geometry of rays giving rise to the glory (neglecting surface-wave contribution). A plane wavefront is incident upon a spherical drop: several ray paths are drawn, one of which is the path abcde. It can be seen that rays adjacent to the central one correspond to a curved emerging wavefront, with a virtual focus at F. Because of cylindrical symmetry, the Agure must be rotated about the axis cs; the outgoing rays (which mutually interfere) deAne a toroidal wavefront from the virtual circular source described by the point F. (Redrawn from [7].)
Bryant and co-workers have carried out experimental and numerical investigations of the glory, and have developed some interesting phenomenological models. In [54] the scattered intensity of the axial rays is given as Ia =
R 2 [(N 2 − 2N + 2) + N (2 − N )cos 4N ] ; 2(2 − N 2 )
(3.9)
where R is the reTection coeYcient at normal incidence. This axial contribution is not suYcient to explain the glory phenomenon [7] so Bryant and Cox [54] use the exact Mie theory to elucidate information about surface wave contributions to the glory. (It should be pointed out that in addition to surface waves there is also the possibility of such waves taking “short cuts” through the sphere; this is addressed below.) In particular they calculated scattered intensities ◦ ◦ at 90 and 180 of incident light polarized both normal to and parallel to the plane of scattering ◦ for a size parameter 200. The total cross section contains large spikes at the 180 intensity; ◦ this feature, much reduced in magnitude, also remains at 90 ; with spikes occurring alternately in one polarization but not the other (see Fig. 2 in [54]). The periodicity of these spikes in is ◦ twice that of the total cross section and the 180 intensity. In the latter there is a sinusoidally varying component of period near 0:8. This “ripple” in the extinction curve (i.e. the ratio of total cross section to geometric cross section vs ) thus becomes enormously ampliAed in the ◦ di9erential cross section at 180 . It is well known that the cross sections are extremely sensitive to small changes in the input parameters [10].
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Mention has been made already of the localization principle: terms of order n in the Mie expansion correspond to a ray passing the origin at a distance (n + 12 )=k because as is known in connection with partial-wave analysis, the nth partial wave is associated with angular momentum M = n(n + 1)˜ ∼ bn = (n + 12 )˜ (the last approximation being valid for large values of n). The appropriate impact parameters are concentrated within the annular region [n =2 ; (n + 12 ) =2 ]; and for the mid-range impact parameter bn ; M = pbn , where p is the incident linear momentum, which in the semiclassical approximation is associated classically with a particle of momentum h= . Hence bn ≈
(n + 12 ) : k
The convergence of the Mie expansion for n& = ka is (physically) a consequence of the fact that the rays no longer interact with the scattering sphere (again, though, this is not the complete picture; as has been noted in the introduction, tunneling to the surface can occur for impact parameters greater than a, giving rise to the sharp resonances found in the ripple: this will ◦ be further elaborated in Section 5). The results for the scattering at 180 indicate that they are determined almost entirely by rays nearly tangential to the surface together with a smaller contribution from the axial rays. Based on these results, an interesting phenomenological model for the surface waves was developed in [54]. The idea is that a tangentially incident bundle of rays is trapped on the spherical surface and propagates around it with constant attenuation due to reradiation. In terms ◦ ◦ ◦ ◦ of unit amplitude at its point of tangential entry at 0 , at 90 it is A; A2 at 180 ; A3 at 270 , ◦ and so on, where |A| ¡ 1: The scattering amplitude at 0 can be represented as ◦
S(0 ) = 5(1 + A4 + A8 + A12 + · · ·) + B ;
(3.10)
where 5 is a real constant, and the term B represents the contribution to zero-degree scattering from other mechanisms. Clearly ◦
S(0 ) = 5(1 − A4 )−1 + B ;
(3.11)
so writing A4 = bei, and using the optical theorem [7] in the form ◦
Q = 4 Re[S(0 )] −2 ; Q being the extinction coeYcient, it is found that 45 1 − b cos , Q = Q0 + 2 ;
1 + b2 − 2b cos ,
(3.12)
(3.13)
where Q0 is determined by the term B. The authors then consider a region of resonance near , = 2m ; m being an integer and , being replaced by , mod 2 : Thus it follows that if b = e−g and , and g are both small Q(,) ≈
D1 + D2 ; 1 + (,=g)2
(3.14)
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where D1 and D2 are insensitive to small changes in ,: The period of this peak is 0:809, so , = 2 =0:809, and the authors At the equation for Q(,) to their numerical results for Q, yielding a value for g = 0:008. This means that b ≈ 0:99, and this represents the attenuation around one circumference of the drop. Being so small, it implies that the peak is an interference maximum involving tens, perhaps even hundreds of circuits by the surface wave. A similar ◦ argument for = 180 yields ◦
S(180 ) = 5A2 (1 − A4 )−1 + B
(3.15) ◦
◦
◦
with a corresponding expression for the intensity |S(180 )|2 ; if the 90 and 270 scattering amplitudes are combined then ◦
◦
S(90 =270 ) = 5A(1 − A2 )−1 + B
(3.16) ◦
and from the deAnition of , it is clear that the resonances at 90 have twice the period that is ◦ exhibited by S(180 ) and Q. In a subsequent comparison of experimental results with Mie theory, Fahlen and Bryant [55] developed a two-parameter scalar model in an attempt to gain insight into the scattering mechanisms involved in the scattering data they acquired. Their model again involved surface waves; such waves transport energy tangential to the interface between two media, and arises when that energy strikes such an interface (between two dielectric media) from the more dense side at or exceeding the critical angle for total reTection (allowing for a limiting process for the initial interaction from the less dense side). It is known from several experiments [56,57] that as a surface wave travels along the interface, it continuously reradiates energy back into the denser medium at the angle of critical refraction. It is also attenuated by “spraying” o9 energy tangentially as it propagates around the drop surface. Part of the energy refracted back into the drop (the short cut component) is reTected internally and the remaining part interferes with the surface wave; this occurs over the course of many revolutions around the drop. For a drop of radius a it is shown that the total optical path length for a ray performing M revolutions around the drop and n jumps (short cuts) across it is l(n; M ) = 2 Ma + a + 2na(tan 5 − 5) ;
(3.17)
where is the scattering angle subtended by both the chord and the arc AB (see Fig. 12). Denoting the amplitude of such a ray by A(n; M ), the probability (statistical weight in [55]) for n jumps by P(n; M ) and the surface wave attenuation by exp(−6z), z being the surface distance deAned by z = 2 Ma + a − 2n5a ; then A(n; M ) = P(n; M ) exp[ − 6z + 2 il(n; M )= ] :
(3.18)
The (complex) total amplitude At is the sum of all possible paths, i.e. ∗
At =
n ∞ M =0 n=0
A(n; M ) ;
(3.19)
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Fig. 12. A general representation of both surface-wave (to A and from B) and “shortcut” components contributing to the glory formed from water droplets. (Redrawn from [55].)
where n∗ 6 (2 M + )=25 must be an integer. If P(n; M ) = exp(−pn) and g = 6 =2 then it may be shown that n∗ ∞ Re At = exp[ − pn − g ( + 2 M − 2n5)] cos[ ( + 2 M + 2n(tan 5 − 5)] M =0 n=0
(3.19a)
with a similar expression (involving a sine term) for Im At : The two parameters in this mathematical model are p (related to the jump probability) and g (related to the attenuation of surface wave energy). By varying them to obtain the best agreement between the experiment and the Mie theory curve for intensity vs size parameter
, Fahlen and Bryant And values p = 0:01 and g = 0:004: For these values the agreement is good: the correct periodicity is present and also the sharp intensity resonances, even though no allowance has been made for polarization. The combination of surface wave propagation and short cuts across the drop are consistent with the more technical theory of Nussenzveig [6] and contain many (but not all) of the essential components for a satisfactory explanation of the glory. As mentioned above in connection with tunneling to the surface, a complete account of the glory has now been a9orded by more recent work of Nussenzveig and coworkers; this will be reviewed in Sections 5.6 and 5.8. 3.2. Rainbow glories The topic of rainbow-enhanced forward and backward glory scattering (rainbow glories) was studied by Langley and Morrell using both the geometrical optics approximation and Mie theory [58] (there are many further useful references in this paper). For certain values of the refractive index N in spherical scattering objects, a higher-order rainbow can coincide with a forward or backward glory to produce exceptionally strong scattering in those directions. The relevant values of N can be found from geometrical optics, but as already seen, in order to characterize the glory adequately it is necessary to consider the e9ects of surface waves, tunneling, internal resonances and high-order rainbows [53,59,60] along with complex angular momentum techniques [61,10] as noted in the previous subsection.
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The total deviation angle D undergone by a ray is, in the present notation, D = 2i − 2rP + (P − 1) :
(3.20)
Forward and backward scattering correspond to D = L ; so from this relation and Snell’s law the glory condition is found to be N = (csc r) cos{rP − (P − L)=2} :
(3.21)
The ray paths may be conveniently designated by the ordered pair (P; L) where P is the number of chords the ray makes inside the sphere and L is the number of times the ray crosses the optical axis. The authors demonstrate that for a given N there may be 0; 1 or 2 glory rays of type (P; L) with the following ranges: 3.2.1. Ray sec(L =2P) ¿ N ¿ 0; csc( =2P) 6 N 6 P;
P − 1¿L ; P − 1=L :
3.2.2. Rays sec(L =2P) 6 N 6 N ∗ ;
P − 1¿L¿0 :
N ∗ is the refractive index for which a rainbow coincides with a glory; using the (necessary) rainbow condition d D= d i = 0 is, on using Snell’s law and Eq. (3.20), tan i = P tan r :
(3.22)
The requirement that both the rainbow and glory conditions be satisAed is therefore P tan r = B(tan rP)B
(3.23)
where B = (−1)P−L−1 : For a given (P; L) the solution of this equation for r when substituted in (3.21) gives the value of N ∗ . A table of eight such values can be found in [58] along with detailed Agures and calculations for the physical optics approximation and the Mie theory; we note several features here. By comparing incremental areas of the wavefront entering and exiting the sphere, the parameter A = 2(P tan r − tan i) occurs, where A is zero at the rainbow angle. Using this parameter, the radius of curvature of the wavefront at the exit plane is deAned to be A Q(N ) = : A + sin i Q = 0 corresponds to a cubic wavefront, but the glory wavefront may be diverging (Q ¿ 0) or converging (Q ¡ 0) at the exit plane, depending on the value of N: Many of the mathematical details follow from the earlier work [62,63]. Normal glory scattering leads to an irradiance with size parameter dependence O( ); for a rainbow-enhanced glory this becomes O( 4=3 ), based on the physical optics approximation, and the Mie theory calculations conArm this at the predicted values of N: The polarization properties of rainbow glories are found to be similar to those for normal glory scattering.
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3.3. The forward glory The optical glory is a strong enhancement in near-backward scattering of light from water droplets with size parameter &100, arising from a combination of almost grazing incident rays, di9raction of and interference between surface waves, ray shortcuts across the drop interior, and complex rays in the shadow ofhigh-order rainbows. The enhancement is associated with an axial focusing e9ect of order G , where G = kb is the glory angular momentum [7], but for water droplets the major contribution to the phenomenon is from complex trajectories. Nussenzveig and Wiscombe [64] studied the forward (optical) glory, and noted the following necessary conditions for such a glory to be observed in Mie scattering: (i) the contributing ray paths should involve the smallest possible number of internal reTections to minimize attenuation by reTection and absorption along the optical path (the latter occurring if Im N ¿ 0); (ii) the incident rays should be nearly tangential to minimize energy losses from internal reTections (so the internal reTectivity should be as large as possible); and (iii) the e9ect must be large enough not to be swamped by the intense forward di9raction peak (which is O( )). The smallest number of internal reTections that can lead to a forward glory is two; this requires Re N ¿ 2. For Re N = 2, tangential incidence gives rise to an internal geometrical resonance path [6] in the shape of an equilateral triangle. For Re N ¡ 2, of interest in this report, the triangle does not close, resulting in an angular gap > to be bridged by surface rays, (where for simplicity of notation we write N for Re N ) [64] > = 6 arcsin(1=N ) − : ◦
(3.24) ◦
For N = 1:33, > 112:5 , but for N = 1:85; > 16 : This is indicative of the sensitivity of >(N ), and as pointed out by Khare [42], for large values of p (p − 1 being the number of internal reTections) this becomes extreme, and care must be taken in employing asymptotic expressions for the Debye amplitudes (note that if > ¡ 0, the contribution is from rainbow terms, not surface waves [53]). In their calculations comparing the exact Mie theory with the complex angular momentum approximation, Nussenzveig and Wiscombe note that the di9erence between the extinction eYciencies does not show the irregular ripple Tuctuations seen in the backward glory, but instead a regular, nearly sinusoidal oscillation arising from interference between the forward di9raction peak and the forward glory contributions. Comparison of the predicted oscillation periods from both methods shows excellent agreement; the complex angular momentum theory predicting that [61] = 2 =(6 N 2 − 1 + >) (3.25) √ Note that for the next geometrical resonance (an inscribed square), occurring for N = 2, the corresponding has a factor of 8 instead of 6 in (3.25), and the corresponding > is given by > = 8 arcsin(1=N ) − 2 :
(3.26) −1=3
Concerning polarization, the authors state that for scattering angles and not 1, both scattering amplitudes tend to be dominated by the Airy pattern of Fraunhofer di9raction by a circular disk, namely J1 ( )= , which gives rise to the forward di9raction peak. The amplitudes are polarized in a similar manner to those in the backward glory: one component
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exhibiting the Airy pattern behavior and the other with a J1 ( ) angular dependence. In short, the forward glory is strongly polarized, with the electric (parallel) component dominating, characteristic of surface waves [61]. 4. Semi-classical and uniform approximation descriptions of scattering In a primitive sense, the semi-classical approach is the “geometric mean” between classical and quantum mechanical descriptions of phenomena; while one wishes to retain the concept of particle trajectories and their individual contributions, there is nevertheless an associated de Broglie wavelength for each particle, so that interference and di9raction e9ects enter the picture. The latter do so via the transition from geometrical optics to wave optics. The differential scattering cross section is related to the quantum scattering amplitude f(k; ) and this in turn is expressible as the familiar partial wave expansion [65]. The formal relationship between this and the classical di9erential cross section is established using the WKB approximation, and the principle of stationary phase is used to evaluate asymptotically a certain phase integral (see [10], Section (1:2) and below for further details). A point of stationary phase can be identiAed with a classical trajectory, but if more than one such point is present (provided they are well separated and of the Arst order) the corresponding expression for |f(k; )|2 will contain interference terms. This is a distinguishing feature of the ‘primitive’ semiclassical formulation, and has signiAcant implications for the four e9ects (rainbow scattering, glory scattering, forward peaking and orbiting) noted in Appendix A. The inAnite intensities (incorrectly!) predicted by geometrical optics at focal points, lines and caustics in general are “breeding grounds” for di9raction e9ects, as are light=shadow boundaries for which geometrical optics predicts Anite discontinuities in intensity. Such e9ects are most signiAcant when the wavelength is comparable with (or larger than) the typical length scale for variation of the physical property of interest (e.g. size of the scattering object). Thus a scattering object with a “sharp” boundary (relative to one wavelength) can give rise to di;ractive scattering phenomena. Under circumstances appropriate to the four critical e9ects noted above, the primitive semiclassical approximation breaks down, and di9raction e9ects cannot be ignored. Although the angular ranges in which such critical e9ects become signiAcant get narrower as the wavelength decreases, the di9erential cross section can oscillate very rapidly and become very large within them. As such the latter are associated with very prominent features and in principle represent important probes of the potential, especially at small distances. The important paper by Ford and Wheeler [66] contained transitional asymptotic approximations to the scattering amplitude in these ‘critical’ angular domains, but they have very narrow domains of validity, and do not match smoothly with neighboring ‘non-critical’ angular domains. It is therefore of considerable importance to seek uniform asymptotic approximations that by deAnition do not su9er from these failings [68]. Fortunately, the problem of plane wave scattering by a homogeneous sphere exhibits all of the critical scattering e9ects (and it can be solved exactly, in principle), and is therefore an ideal laboratory in which to test both the eYcacy and accuracy of the various approximations. Furthermore, it has relevance to both quantum mechanics (as a square well or barrier problem) and optics (Mie scattering); indeed, it also serves as a model for the scattering
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of acoustic and elastic waves, and was studied in the early 20th century as a model for the di9raction of radio waves around the surface of the earth [67]. The existence of a rainbow angle, R (or 0 as has also been used above) is not just relevant to the meteorological phenomenon; along with the glory it has counterparts in atomic, molecular, heavy-ion and nuclear scattering (though the glory is usually a forward glory in these situations). These fundamentally important aspects of ‘scattering problems’ will be summarized in Section 7, but a few comments are in order at this juncture. In general, whenever a plane wave is scattered by a spherically symmetric potential there is an associated scattering angle R (measured from the forward direction) for which the di9erential cross section /() is large [68]. Just as with the optical rainbow, on the “shadow” side of R the scattering drops rapidly to zero; on the “lit” side /() oscillates. Ford and Wheeler [66] referred to this as rainbow scattering whenever it occurs in other scattering situations, where provided R = 0; it is called the rainbow angle. In particular rainbow scattering occurs in the intermolecular collisions arising during molecular beam experiments, where “molecular rainbows” can provide information about the intermolecular forces ([20,69]; an early non-mathematical account can be found in [70]); this is often referred to as an inverse problem in the mathematics literature. The inverse problem in scattering theory refers to the construction of the potential directly from the measured data, e.g. in molecular scattering, a set of phase shifts may be obtained from the observed angular and energy dependence of the cross section, and the potential is deduced from these phase shifts. There is an enormous amount of literature on the subject of inverse problems in general, ranging from the mathematical theory (e.g. questions of existence and uniqueness of the potential) to the practical aspects of implementing the theory. Since uniqeness is not usually guaranteed in problems of this type, there has been interest in identifying all equivalent potentials for a given set of phase shifts. For a summary of both aspects of the problem prior to 1974 (but with emphasis on practical inversion procedures) see the excellent review by Buck [71] where many further references can be found. A discussion of more recent developments in the Aeld is found in Section 6.5. The types of phenomena identiAed thus far typically arise when the scattering “center” is much larger than the wavelength of the incident beam (as in the meteorological rainbow); under these circumstances a large number of partial waves contribute to the scattering, giving rise to signiAcant variations in the di9erential cross section. Under these conditions, as pointed out by Berry [68], the system is close to both the geometrical optics limit of electromagnetism and the classical limit of quantum mechanics. However, the calculations are not the same for optical and atomic=molecular=nuclear rainbows because in the former case the refractive index is discontinuous (or at least changes signiAcantly over distances much smaller than a wavelength, i.e. at the drop surface) and in the latter case the potential is continuous (this latter point has been made already in connection with truncated potentials). There are several approximations that have been used to describe rainbow scattering (to be understood in the general sense in this section). In [68] these are classiAed as (i) the classical approximation, (ii) the “crude semi-classical” (or WKB) approximation, (iii) the Airy approximation, and (iv) the uniform approximation. Mention has been already made of these, but now more details will be included. The classical approximation can be obtained by smoothing out the oscillations in the asymptotic form of the exact cross section for angles far from R and analytically continuing the result to angles near R ; this is basically equivalent to D7escartes’ theory in the case of the
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optical rainbow, and as in that case, it diverges at R : The ‘crude semi-classical’ approach is the unsmoothed asymptotic form of /() far from R and it also diverges if continued analytically to R : The Airy approximation here means the application to continuous potentials of the previously discussed account of the light intensity in the neighborhood of a caustic and developed by Airy for the optical rainbow problem. This gives the form of /() close to R but rapidly becomes inaccurate as deviates from R by more than a degree or two. Since it is such an important paper, we will defer discussion of the Ford=Wheeler contribution [66] until later, and content ourselves with summarizing here the seminal contribution of Berry [68] to the fourth classiAcation—the uniform approximation—which in some respects is very similar to (and pre-dates) the application of the Chester et al. [72] saddle-point method to the rainbow problem. In his valuable paper Berry discusses the reason for the existence of all these di9erent approximations. Each is valid in a restricted angular domain, and the asymptotic expansion of the scattering amplitude f() with respect to the (very) small parameter ˜ (when | − R | is large) changes its form from a series in powers of ˜1=2 to a series in powers of ˜1=3 (when | − R |1). This phenomenon is familiar for functions deAned by ordinary di9erential equations, which is why the term ‘crude semi-classical’ is associated with the WKB approximation. By contrast, the Airy approximation is called a transitional approximation: it ‘heals the wound’ between one region and another in a smooth, well-deAned manner [73]. The uniform approximation, as its name implies, is valid for the whole variable domain (see [74,75]). Berry transforms the eigenfunction expansion of f() by the Poisson sum formula (see Appendix C) to give a series of integrals (as does Rubinow in an important but infrequently noted paper [76], and Nussenzveig [33]; indeed, in 1950 Pekeris [77] suggested that the Poisson formula is the natural mathematical tool to use when considering an eigenfunction expansion in a region where ray=classical concepts are appropriate). As noted earlier in connection with the Mie series, such a transformation is exceedingly helpful in practical terms because the standard Faxen–Holtsmark formula for f(), while exact, converges extremely slowly for in the neighborhood of R ; often many thousand partial waves are required in this region. The application of the Poisson sum formula allows f() to be written in terms of integrals which, in the notation of [68], may be referred to as Im± (though we do not write them down here; see Eqs. (4)–(9) in [68]). The integrands of Im± are rapidly oscillating functions, and it is known that the main contributions to the integrals come from the neighborhoods of stationary points on the positive real axis (when they exist) [78]. We will describe the basic features of this analysis below, but it suYces here to note that the points of stationary phase are deAned in terms of a quantity @(l) (this is in fact the classical deTection function—see below and Appendix A); rainbow scattering arises whenever @(l) has an extremum. As we will see in Section 5, there are in fact two real stationary (saddle) points if ¡ R ; if ¿ R they are complex conjugates, having moved away from the real axis, and only one of these contributes signiAcantly to the amplitude. At = R they coalesce into a third-order saddle point, for which the semiclassical approach is not valid (recall that the direction R deAnes a caustic direction). The uniform approximation, based on the work of Chester et al. [72] comes, so to speak, to the rescue. Instead of treating the saddle points separately, as in semi-classical methods, or as essentially coincident, as in the Airy approximation, this method maps their exact behavior onto that of the stationary points of the integrand in the Airy function. The resulting equations
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for the rainbow cross section /R are rather complicated, but Berry shows that on both the lit and shadow sides of R they reduce in the appropriate limits to that deduced from the Airy approximation. Ford and Wheeler [66] identify four mathematical approximations that can be considered to deAne the term “semi-classical” approximation. The di9erential scattering cross section into unit solid angle at is /() = |f()|2 ;
(4.1)
where f() is expressed in the form ∞ 1 f() = (2l + 1)(e2i%l − 1)Pl (cos ) ; 2ik
(4.2)
l=0
%l being the phase shift for the lth partial wave. The Arst approximation is: (i) %l is replaced by its WKB approximate value, namely
∞ l − r0 + (A(r) − k) d r ; %l = + 4 2 r0 where
2m(E − V ) (l + 12 )2 A(r) = − ˜2 r2
(4.3)
1=2
(4.4)
and r0 is the turning point of the (classical) motion, deAned by A(r) = 0 (see Appendix A). Physically, this approximation is equivalent to the requirement that the potential V be “slowly varying”, i.e. 1 dV (4.5) kV d r 1 : As Ford and Wheeler point out, the most important property of the WKB phase shift is its simple relation to the classical deTection function @(l): d% @(l) = 2 l : (4.6) dl This allows a correspondence of sorts to be made between the quantum and classical results. (ii) The second approximation concerns the replacement of the Legendre polynomials by their aymptotic forms for large values of l: Thus (a) ∼ [ 1 (l + 1 ) sin ]−1=2 sin (l + 1 ) + ; sin & l−1 Pl (cos ) = (4.7) 2 2 2 4 and (b) ∼ (cos )l J0 [(l + 1 )]; Pl (cos ) = 2
sin . l−1 :
(4.8)
For this approximation to be valid, many l-values must contribute to the scattering at a given angle (and the major contribution to the scattering amplitude comes from values of l1). The third approximation is
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(iii) Replacement of the sum over l by an integral with respect to the same variable. Again, this means that many partial waves should contribute to the scattering and also that %l (l) should vary slowly and smoothly. Approximations (i) and (ii) render both %l and Pl continuously di9erentiable functions of their arguments (in general), so that (iii) is appropriate. Approximations (i), (ii)(a) and (iii) (for angles not close to 0 or ) yield the semi-classical form of the scattering amplitude
∞ −1 1=2 fsc = − k (2 sin ) (l + 12 )1=2 [ei,+ − ei,− ] d l ; (4.9) 0
where the phase functions ,± (l; ) are deAned by ,± (l; ) = 2%l ± (l + 12 ) ±
4
(4.10)
and the result l (2l + 1)Pl = 0 has been used [79]. A fourth approximation, not always necessary, is that (iv) It may be necessary for the integral in (iii) above to be evaluated by the method of stationary phase, or the method of steepest descent, as we have seen already. Mott and Massey [79] examined the integral under circumstances such that there is only one point of stationary phase (corresponding to @(l) varying monotonically between 0 and ± ). This is not the case in rainbow or glory scattering, where @(l) is not monotone everywhere in its domain. Ford and Wheeler study Ave special features of scattering in the semiclassical approximation; we mention only the interference=rainbow=glory features in keeping with the emphasis of this review. The mathematical details can be found in [79]. Because (classically) several incident angular momenta may correspond to the same scattering angle at a given energy, the total cross section will in general be the sum of di9erent contributions from the di9erent branches; and when these are well separated, each will give an independent contribution to the scattering amplitude which may be evaluated by the method of stationary phase. Then, formally, the semi-classical amplitude becomes fsc = (/cl )j1=2 ei j ; (4.11) j
where j is a phase angle for the jth contributing branch. If j = 1; 2 only, then 1=2 i( 2 − 1 ) 2 /sc = |(/cl )1=2 | 1 + (/cl )2 e
(4.12)
so that /sc will oscillate between minimum and maximum values deAned by the quantities 1=2 2 [(/cl )1=2 1 ± (/cl )2 ] , with “wavelength” ] = 2 = |l2 − l1 |, the l1; 2 being the points of stationary phase. The rainbow angle corresponds to the singularity arising in the classical cross section when d @(l)= d l vanishes (the cross section contains a factor (d @(l)= d l)−1 ). Near a rainbow angle, @(l) may be approximated by the quadratic function @(l) = R + q(l − lR )2 ;
(4.13)
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where the terms have obvious meanings. On the “lit” or bright side of the rainbow angle the classical intensity is
lR + 12 /cl = |q( − R )|−1=2 (4.14) k 2 sin R and on the “dark” or shadow side, the classical intensity is zero (if there are no additional contributing branches of the deTection function @(l)). This of course is identical with the predictions of geometrical optics for the optical rainbow. In terms of the phase shift %l however, %l = %R ± (R =2)(l − lR ) + (q=6)(l − lR )3 ;
(4.15)
where the ± is necessary for the antisymmetry (see Eq. (4.10) for ,± above). For R ¿ 0 the dominant contribution to integral (4.9) will come from the term containing the factor exp(i,− ); this yields essentially the same functional form for the scattering di9erential cross section as does the original Airy theory for the intensity of the optical rainbow. Thus fsc = [2 (lR + 1=2)=k 2 sin ]1=2 q−1=3 ei Ai(x)
(4.16)
where x = q−1=3 (R − ) ; is a measure of the deviation from the rainbow angle, and = 25R − =4 + (lR + 1=2)(R − )
(4.17)
for @(lR ) ¿ 0: If @(lR ) ¡ 0, then R − must be replaced by its negative in the expressions for x and . The quantity 5R is the intercept of the tangent line to the (%R ; lR + 1=2) curve with the vertical axis, i.e. 5R = [%l − (l + 1=2)(d %l = d l)]l=lR (see Fig. 1 in [66]). The form of the Airy integral used there is
1 ∞ 1 3 Ai(x) = exp ixu + iu d u : 2 −∞ 3
(4.18)
(4.19)
The di9erential cross section near the rainbow angle then has the form /sc = [2 (lR + 1=2)=k 2 sin ]|q−2=3 |Ai2 (x) :
(4.20)
If other branches of @(l) do contribute at R , then fsc above must be combined with the other contributing amplitudes before the absolute square is taken (this in fact arises (theoretically) for magnetic pole scattering, and also for ion-atom scattering (see a detailed discussion in [66])). Provided 0 ¡ @(l) ¡ , or − ¡ @(l) ¡ 0; /sc can be described entirely in terms of the classical cross section, together with interference e9ects (not discussed here) and rainbow scattering. If however @(l) passes smoothly through 0; ± ; etc. then the vanishing of sin @(l) for l; |d @= d l| ¡ ∞ leads to a singularity in the cross section for both forward and backward
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scattering. The optical=meteorological terminology is again borrowed for this situation: it is referred to as a glory. In the case e.g. of a backward glory, near @(l) = , we may write the approximate form @(l) = + a(l − lg )
(4.21)
and then /cl in the backward direction is just the sum of two equal contributions from @ ¡ and @ ¿ , i.e. /cl =
2lg : k 2 |a|( − )
(4.22)
From (4.21) the phase shift, for those values of l that contribute most to the glory, may be written as a %l = %g + (l − lg ) + (l − lg )2 (4.23) 2 4 and after some rearrangements of the integral in the expression for fsc , it can be shown that the glory cross section is /sc = (lg + 1=2)2 (2 =k 2 |a|)J02 (lg sin ) ;
(4.24)
when there are no interference e9ects. For a forward glory the result is similar; there is a di9erent phase term but this makes no contribution of course to /sc : Thus the singularity in /cl is replaced by a Anite peak in both forward and backward directions of max = (lg + 1=2)2 (2 =k 2 |a|) : /sc
(4.25)
The Bessel function oscillations may be interpreted as resulting from interference between contributions from the two branches of @(l) near a glory, i.e. @ ¿ (or 0) and @ ¡ (or 0). Ford and Wheeler point out that when the intensity is averaged over several such oscillations (using the result J02 (x) = (1= x)) then /sc reduces to the classical expression /cl : For further early references to atomic and molecular scattering, see [80 –87]; more recent work in these contexts and in nuclear scattering is reviewed in Section 6.5.
5. The complex angular momentum theory: scalar problem 5.1. The quantum mechanical connection An important summary of the main results for high frequency scattering by a transparent sphere has been provided by Nussenzweig [34]. A rather comprehensive set of references to this and related work up to 1991 can be found in his book [10]; there are several more recent papers, along with those of co-workers (e.g. [88,89]; see also an early paper on square well and barrier potentials [90]).
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As has been pointed out already, in terms of the size parameter = ka, the wavenumber k; the real refractive index N and the impact parameter for the lth partial wave bl =
(l + 1=2) ; k
(5.1)
there are certain assumptions necessary for reasonable approximations to be made to the scattering problem. The presence of absorption can be incorporated by allowing N to be complexvalued, but that will not be necessary here; although according to [5] it should not be unduly diYcult to carry out. The essential mathematical problem for scalar waves can be thought of either in classical terms, e.g. the scattering of sound waves, or in wave-mechanical terms, e.g. the non-relativistic scattering of particles with momentum p = ˜k by a square potential well (or barrier) of radius a and depth (or height) V0 , i.e. (2)
V (r) = − V0 V (r) = 0
(0 6 r 6 a) ;
(r ¿ a) :
The governing time-independent radial SchrRodinger equation for this potential is 2 d l(l + 1) 2m − + 2 (E − V (r)) (r) = 0 ; dr 2 r2 ˜
(5.2)
(5.3)
where (r) is the scalar radial wavefunction and the third term represents the centrifugal barrier [91] and E = p2 =2m is the eigenvalue parameter (energy: note that expressed in units for which ˜ = 2m = 1; E is merely equal to k 2 ). In the l = 0 case in particular, this equation has a useful “classical” form if the wavenumber k is used, i.e. 2 d 2 2 + k N (r) (r) = 0 ; (5.4) dr 2 where the refractive index is, for the potential well [5,76,92] 2mV0 1=2 N = 1+ 2 2 ; ˜ k
(5.5)
m being the mass of the “particle”. It follows that N ¿ 1 corresponds, for Axed k (or energy), to a potential well (crudely, the raindrop can “trap” the ‘probability waves’ to some extent) while N ¡ 1 corresponds to a barrier (inhibiting the waves to a lesser degree, as with an air bubble in water). It is the former case that will concern us here. Other related interpretations are possible: for a Axed V0 , N is dispersive, being frequency-dependent, and if N is Axed then the potential is proportional to the energy, as noted earlier. Partial waves for which bl . a (i.e. l+ 12 . = ka) are substantially distorted by the spherical scatterer, so that at least terms are required for reasonable accuracy (in fact it transpires that a better measure of the number of terms is given by l+ ∼ + c 1=3 ; where c&3; [93]). This can be interpreted in terms of the centrifugal barrier penetration: from the radial equation (5.3)
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the e9ective potential is (see Appendix A) U (r) = V (r) +
˜2 l(l + 1)
: (5.6) 2mr 2 In fact, if the WKB approximation is to be applied, a correction is in order, as pointed out by Berry [94]; the term l(l + 1) should be replaced by (l + 12 )2 : This is because the ordinary WKB solutions are not valid near the origin (because U (r) varies rapidly there) and so cannot satisfy the boundary condition that the reduced radial wavefunction is zero at the origin. However, as shown by Langer [74] and Bertocchi et al. [95], the WKB approximation is valid near the origin provided the above change is made in the e9ective potential U (r): Under these circumstances the modiAed WKB wavefunction is r exp{±i A(r) d r } (r) ˙ ; (5.7) {A(r)}1=2 where from (4.4) 2m A2 (r) = 2 ˜
E − V (r) −
˜2 (l + 12 )2
2mr 2
:
(5.8)
Note also that U (r) is discontinuous at r = a, introducing a barrier, so bl ¿ a corresponds to an energy level below the top of the centrifugal barrier. Berry states that the transmissivity of the barrier up to r = a (from below) is 4N Tl = 2 exp(2 l ) : (5.9) N +1 From where does this particular multiplicative factor come? The transmissivity T for a stepbarrier of heights V1 and −V0 with barrier thickness a` (see Fig. 13) is obtained in the usual manner by matching the wavefunction and its Arst derivative, and is found to be [65] (recall that, as deAned above, V0 ¡ 0 for a barrier): −1 A K 2 K 2 K 2 2 1+ cosh Aa` + − sinh Aa` ; (5.10) T =4 k k k A where ˜2 k 2
˜2 K 2
˜2 A 2
= V1 − E : 2m 2m 2m The centrifugal barrier is, to a Arst approximation, a potential of this type; speciAcally it is extremely narrow because the high l-value imposes a steep decline beyond r = a. This of course means that a`1 and hence sinh2 Aa`1, cosh2 Aa` ≈ 1 and under these circumstances 4N T= ; (5.11) (1 + N )2 where
= E + V0 ;
V0 1=2 k N = = 1+ K E
=E
and
(5.12)
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Fig. 13. The location of the “edge domain” (rays passing close to the “edge” of the sphere), deAned by
− c 1=3 = l− . l . l+ = + c 1=3 for the lth partial wave. The constant c is of order one. The potential U (r; ) is the sum of the centrifugal barrier term and the square-well potential (see Eqs. (5.6), (A.5) and [47]). The angular momentum = l + 12 : In the edge domain the energy associated with these is, as shown, within a small neighborhood of the top of the barrier.
(There appears to be a small typographical error in [94], perpetuated in [6], assuming this potential is the basis for the statement in the former paper.) In practice, of course, the centrifugal “spike” barrier varies with energy, so this expression for T has to be modiAed by the “tunneling factor”, deAned below, which takes this into account. At the other extreme, for a thick barrier, we And that the transmissivity is dominated by the exponential term exp(−2Aa) ` ;
which rapidly approaches zero as a` increases. The expression for Tl exploits the optical=quantum analogy: the refractive index N being the ratio k=K as noted above. The phase integral l is deAned in terms of the turning points at r = a and r = r1 , i.e. for which A(a) = 0 = A(r1 ). Thus
r1 |A(r)| d r (5.13) l=− a
from which it follows that, in the interval (a; r1 ); A2 = k 2 −
(l + 12 )2 ¡0 : r2
In terms of the variable x = kr the phase integral can be written as 1=2
l+1=2 dx 1 2 2 l+ −x : l=− 2 x
(5.14)
(5.15)
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In particular, near the top of the barrier this integral may be approximated straightforwardly using, for example, the Trapezoidal Rule. For just two trapezoids
b b−a 1 1 f(x) d x ≈ f( ) + 2f + l + +f l+ : (5.16) 4 2 2 a After some algebra this yields the expression √ 3=2 1 3 2 1=3 l+ −
: (5.17) l ≈− 4 2 √ (In [6] the numerical factor is given as −2 2=3, the reciprocal of the value above: it is not clear if this is merely a typographical error or due to a di9erent method of approximating the integral; in either case it is close to unity and does not alter the conclusions that follow.) From Eq. (5.9) it follows that the transmissivity for impact parameters bl exceeding a is signiAcant only when l ≈ O(1), i.e. e9ectively in the range ¡ l ¡ l+ (see Fig. 13). 5.2. The poles of the scattering matrix The dimensionless scattering amplitude f(k; ) is expressed as a partial-wave expansion ∞ 1 f(k; ) = (ika)−1 l+ [Sl (k) − 1]Pl (cos ) ; (5.18) 2 l=0
Sl being the scattering matrix element for a given l and Pl being a Legendre polynomial of degree l: In [90] the expression for Sl has been derived for a (spherical) square well=barrier: for the former this is (2) ln hl ( ) − N ln jl (5) h(2) l ( ) Sl = − (1) ; (5.19) hl ( ) ln h(1) ( ) − N ln j (5) l l where using Nussenzveig’s notation, ln represents the logarithmic derivative operator, jl and hl are spherical Bessel and Hankel functions respectively; = ka; as noted earlier, is the dimensionless external wavenumber, and 5 = N is the corresponding internal wavenumber [5]. Poisson’s sum formula in the form (see Appendix C)
∞ ∞ ∞ 1 m f l+ = (−1) f( )e2im d (5.20) 2 0 m=−∞ l=0
is next used to rewrite (5.18) as
∞ ∞ i m f( ; ) = (−1) [1 − S( ; )]P −1=2 (cos )e2im d ;
m=−∞ 0 where, in terms now of cylindrical Bessel and Hankel functions [10], H (2) ( ) ln H (2) ( ) − N ln J (5) S( ; ) = − (2) : H ( ) ln H (1) ( ) − N ln J (5)
(5.21)
(5.22)
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For Axed ; S( ; ) is a meromorphic function of the complex variable , and in particular in what follows, it is the poles of this function that are of interest. They are deAned by the condition (1)
ln H ( ) = N ln J (5) :
(5.23)
In view of relationship (5.5) between the refractive index N and the square well depth −V0 , these zeros may be identiAed with the Regge poles for a square potential well (N ¿ 1) or barrier (N ¡ 1). Several studies have been made of such poles for these potentials [96 –100], both for N ¿ 1 and N ¡ 1: Nussenzveig [90] carried out a detailed investigation of the distribution of such poles, particularly for the case of 1; some of this material is relevant to the rainbow scattering problem so a brief summary of the pertinent details will be provided here for N ¿ 1; but as is noted in [5], the physical interpretation is especially simple for N 1: This implies that 5 1, and renders it appropriate to use the asymptotic expansions for the Bessel and Hankel functions. Physically, this corresponds to an optically dense material or an extremely deep potential well (with, therefore, many energy levels). Mathematical details are provided in [98,99]. Under these approximations the complex -plane is subdivided into seven regions, though the Regge poles themselves can be associated with (i) broad resonances, (ii) narrow resonances and (iii) surface waves. Firstly, there exists a series of poles very close to the real axis; optically they correspond to so-called free modes of vibration of a dielectric sphere. The associated high internal reTectivity (since N is large) and high centrifugal barrier (since l is large) guarantee long lifetimes: the resonance appears when the pole is in close proximity to a physical value of , i.e. = l + 12 : Quantum mechanically, a deep well surrounded by a high barrier gives rise to sharp resonances. All these poles have their counterparts in the - (or k-) plane of course, from which the penetration factors exp(2 l ) and resonance widths Fn can be established. Another class of poles exists, this time not so close to the real -axis, but with almost constant imaginary part. These poles are associated with broad resonances above the top of the centrifugal barrier; again in the -plane the location of the corresponding poles provides information on the penetration factors and resonance widths. A third set of poles, this time near = ; have an extremely important physical interpretation, and to clarify this, we write down the approximation for these, namely n ≈ +
x n ei =3 i + 2 ; 1=3 (2= ) (N − 1)1=2
(5.24)
where −x n is the nth zero of the Airy function Ai(z), i.e. Ai(−x n ) = 0: The Arst and second terms occur in the expression for Regge poles for an impenetrable sphere [10,30], so the poles are related to surface waves. The second term contains a damping contribution which depends on the radius of curvature, and represents damping due to propagation along a curved surface. The Anal term represents damping due to refraction of the surface waves into the sphere, and by virtue of the temporary assumption that N 1, this term is small, so the surface wave damping is determined primarily by geometry rather than refraction. For even larger values of | | there are poles which continue to be associated with surface waves by virtue of their similarity to those
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Fig. 14. The Regge poles of the scattering function S( ; ) for 5 = N 1 and 1 illustrated schematically. The broad and narrow resonances are referred to as Class I poles in [5], and the surface waves are Class II poles. (Redrawn from [5].)
arising in the impenetrable sphere problem. Schematically these are shown in Fig. 14; Nussenzveig points out that they fall into two well-di9erentiated classes: those located near the real -axis (broad and narrow resonances), and those which lie on curves almost orthogonal to the Arst class in quadrants I and III (surface waves). These poles are not surprisingly referred to as Class I and Class II poles. In [99] Class I poles are called “physical” because of their correspondence, for certain values of energy, with the stationary states of a potential well. They behave somewhat similarly to Regge trajectories for Yukawa-type potentials [100]. For very deep wells those poles with large positive real parts move along the real axis at negative energies (giving rise to bound states) and move o9 the axis into the Arst quadrant, at positive energies (giving rise to resonances). At Anite energy, the number of Class I poles in the right half-plane is bounded above. Physically, they are associated with the potential interior, i.e. for r ¡ a, which is why they resemble Yukawa-type Regge trajectories. Class II poles are, by contrast, referred to as “unphysical” in [99]. Even at Anite energies the inAnite number of them in the Arst quadrant have unbounded real parts; as → 0 they all move towards the origin. The trajectories of both classes of poles are brieTy described in [99]. Physically, as we have seen, they are associated with surface waves, and are therefore almost completely determined by the geometrical shape of the scatterer. These contrasting pole behaviors provide a clue as to why the scattering amplitudes for Anite-ranged potentials are so very di9erent in analytic behavior from those potentials with tails extending to inAnity (e.g. Yukawa-type potentials, or integral superpositions thereof, of the form r −1 exp(−5r)). Physically, it seems reasonable to suppose that by “cutting o9” such a potential suYciently far from the origin, the properties so obtained will di9er insigniAcantly (both mathematically and physically) from those without compact support (i.e. inAnite range potentials). However, the above discussion indicates that such a truncation induces the presence of surface waves: the presence now of an in?nite set of Class II poles in addition to the Anite set of Class I poles gives rise to an essential singularity (at Anite energy) at inAnity in the momentum transfer plane, so that the usual (Mandelstam) representation [92] is no longer valid. Even a suYciently rapid exponential decrease in the potential is unable to mimic the cut-o9 potential (i.e. one
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with compact support); the phrase ‘suYciently rapid’ implies a length scale much shorter than the wavelength under consideration, and this is not true at ‘suYciently high’ energies, whereas cut-o9 potentials can support surface waves at arbitrarily high energy. 5.3. The Debye expansion The Watson transformation is applicable to the problem of scattering by an impenetrable sphere; this is discussed in detail elsewhere [30]; see also [10] and Appendix C. However, in the case of a transparent sphere, a large number of Regge poles lie in the neighborhood of the real axis, so if expression (5.21) for f( ; ) were reduced to rapidly convergent contour integrals plus a series of residues from the Regge poles, such a series would of necessity converge very slowly (physically, the di9erence between the two cases arises because wave penetration occurs in the transparent sphere). The number of such poles is of the order of (N − 1) , as may be seen from Eq. (2:37) in [6], which means that the minimum number of residue terms required is at least of the same order as the number of terms needed in the original partial-wave series (a large number of such waves can be near resonance at high frequencies). In [33], the contour integrals were evaluated by the saddle point method, the contributions from which correspond to those arising from geometrical optics, at least to a Arst approximation. This is perfectly acceptable for the impenetrable sphere, for which only direct reTection occurs, but in the present case an incident ray is partially reTected and partially transmitted, the latter in turn being partially (internally) reTected and partially transmitted, and so on. Thus an inAnite series of multiple internal reTections (or interactions with the surface) corresponds to an inAnite number of saddle points, which must be appropriately superimposed to enable the external solution to be constructed. The Debye series is just this: a representation of the scattering problem in terms of surface interactions; Debye adopted this method for a circular cylinder [101], and later van der Pol and Bremmer applied it to the sphere [30,31]. This is done by means of a related but Actitious problem; i.e. by regarding the surface r = a as an interface between two unbounded homogeneous media and matching the solutions to the (radial) equation to determine the internal and external spherical reTection and transmission coeYcients. The interaction of an incoming spherical wave with the sphere is then described in terms of a sequence of surface interactions (partial reTections and transmissions) with interior propagation occurring in an alternate fashion. In particular, the external and internal spherical reTection coeYcients R22 and R11 are given by Eqs. (5.25) and (5.26) for the scalar scattering problem (for the full electromagnetic problem there is a corresponding coeYcient for each polarization; see [10,102] for details):
(2) (2) ln H ( ) − N ln H (5) R22 ( ; ) = − (5.25) (1) (2) ln H ( ) − N ln H (5) and
R11 ( ; ) = −
(1)
(1)
(1)
(2)
ln H ( ) − N ln H (5)
ln H ( ) − N ln H (5)
:
(5.26)
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From conservation of energy (or probability in the quantum–mechanical context) the following relationships hold: 2 H (2) ( ) T21 ( ; ) = 1 |R22 ( ; )|2 + (2) (5.27) H (5)
and
2 H (1) (5) T12 ( ; ) = 1 ; |R11 ( ; )|2 + (1) H ( )
(5.28)
where Tij is the spherical transmission coeYcient from medium i to medium j (i; j = 1; 2; the interior and exterior of the sphere are denoted by indices 1 and 2, respectively). In the limit as the radius a of the sphere tends to inAnity, the above four coeYcients reduce to the standard Fresnel coeYcients for reTection and transmission for a plane interface at normal incidence, i.e. N −1 2N R11 → ; T12 → (5.29) N +1 N +1 and R22 = − R11 ; T21 = T12 : The scattering function S( ; ) (Eq. (5.22)) is to be expressed in terms of surface interactions; in order to accomplish this the external reTection coeYcient R22 is subtracted from S; after some rearrangement this can be written in terms of the quantity. G( ; ) = as S( ; ) =
H (1) (5)
H (2) (5) H (2) ( )
H (1) ( )
R11 ( ; ) R22 ( ; ) + T21 ( ; )T12 ( ; )
(5.30)
∞ H (1) (5)
H (2) (5) p=1
[G( ; )]p−1 :
(5.31)
This is the celebrated Debye expansion, arrived at by expanding the expression [1 − G( ; )]−1 as an inAnite geometric series. The Arst term inside the parentheses represents direct reTection from the surface; the pth term represents transmission into the sphere (from the term T21 ) subsequently “bouncing” back and forth between r = a and 0 a total of p times (from the ratio of Hankel functions preceding the summation), with p − 1 internal reTections at the surface (via the R11 term in G; the case p = 1 corresponds to direct transmission). The Anal factor in the second term, T12 , corresponds to transmission to the outside medium. The multiplicative factor outside the parentheses is phase term due to the surface interaction taking place at r = a rather than r = 0: The origin behaves as a perfectly reTecting “boundary” because of the regularity of the wavefunction there. In summary, therefore, the pth term of the Debye expansion represents the e;ect of p + 1 surface interactions. Regarding convergence of the expansion, it follows from Eqs. (5.28) and (5.30) for real that |G( ; )| = |R11 ( ; )| ¡ 1 :
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Nussenzveig [6] also shows that lim |G( ; )| = 1 ;
(5.32)
→ ±∞
so that any improper integrals for ∈ (0; ∞) must be interpreted in the limiting sense of ∈ (0; $); $ → ∞: In this spirit we may write f( ; ) = f0 ( ; ) +
∞
fp ( ; ) ;
(5.33)
p=1
where f0 ( ; ) =
∞ i
(−1)m
m=−∞
fp ( ; ) = −
m
(−1)
m=−∞
0
and ∞ i
∞
0
1−
∞
H (2) ( ) H (1) ( )
R22 P −1=2 (cos ) exp(2im ) d
U ( ; )[G( ; )]p−1 P −1=2 (cos ) exp(2im ) d
(5.34)
(5.35)
for p ¿ 1: In this last expression U ( ; ) = T21 ( ; )
H (1) (5) H (2) ( ) H (2) (5) H (1) ( )
T12 ( ; ) = U (− ; ) :
(5.36)
In practice, contributions to the integrals in (5.33) for large values of become vanishingly small, so the e9ect of condition (5.32) is negligible. Alternative but equivalent representations to f( ; ) are also possible (see [5] for details). While convergence of the Debye expansion is a fundamental mathematical property, once established it is necessary to ask the practical question: how rapid is the convergence? The earlier expression (5.21) for f( ; ) involves a slowly converging residue series because of the existence of many Regge poles close to the real axis. What happens when the modiAed Watson transformation is applied now, i.e. how are the poles for each term distributed in the complex- plane? A distinct but related question concerns how fast the Debye expansion itself converges. The poles for the Debye expansion di9er from the Regge poles associated with expression (5.23); they are deAned by (1)
(2)
ln H ( ) = N ln H (5)
(5.37)
corresponding to the replacement of standing waves (Regge poles) by travelling waves (Debye interactions). Interestingly, the locations of the poles are the same for each term, but their order varies: they are of order p + 1 for the pth term (p = 0; 1; 2; : : :). There are now two classes of poles to consider, each being symmetrically located with respect to the origin, denoted in [5] by n and n respectively. In the right half-plane, for N | − 1| 1=3 ; using earlier notation n ≈ + ei =3 (x n =6) − i=M
(5.38)
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for N ¿ 1; (the only situation which concerns us in this paper). Similarly, n ≈ 5 + e−i =3 N 1=3 (x n =6) + N=M
(5.39)
for N ¿ 1: The set { n } di9ers little from that corresponding to the impenetrable sphere problem, so it is closely associated with surface waves. This is actually the case for the set { n } as well: the n lie in the fourth quadrant, and arise from the Debye expansion; however, for N ¿ 1 their contributions are negligible (which is not the case for N ¡ 1) and so they will not be discussed further here. In general, when the modiAed Watson transformation is applied to each term in the Debye expansion, the dominant contributions to the asymptotic behavior of each term will be from saddle points and residue series, the former corresponding to rays in geometrical optics. Thus for each term there is a Anite number of saddle points, though this number does increase for each successive term. The residue-series terms converge rapidly (the imaginary part of n increases rapidly with n), and so overall a rapidly convergent asymptotic expansion for each term of the Debye expansion can be obtained. Returning to the question of convergence of the Debye series itself, the saddle-point contributions converge at a rate determined by the damping produced at each internal reTection. This in turn is determined by the Fresnel reTection coeYcient at the interface, and that depends on the impact parameter and the value of N: For most directions (excluding the cases N 1 and N 1) this coeYcient is small and as a consequence the Debye series converges quite rapidly. The convergence rate will be enhanced if the scattering sphere is not transparent, i.e. if absorption occurs in the interior; mathematically this can be accommodated by allowing N to be a complex number. Van de Hulst [7] has estimated that for water droplets (N ≈ 1:33 for orange light) more than 98:5% of the total intensity of incident radiation is accounted for by the rays 1 ; 2 and 3 in Fig. 1(c); these correspond to the Arst three terms in the Debye expansion. Thus the remainder, less than 1.5%, must be distributed among the higher-order terms and the contributions from the residue series. This is not to say, however, that they do not contribute signiAcantly; they can be concentrated within narrow angular domains about certain directions (see Section 5.4 below). Indeed, the phenomenon of the “glory” (see Section 3 and [47]) is a consequence of these residue-series contributions dominating the geometrical ray contributions in a particular angular region. These contributions in general converge considerably more slowly than their saddle-point counterparts. This can be appreciated on physical grounds by considering what happens to the incident ray as the impact parameter increases. Under these circumstances the reTection coeYcient increases towards unity (total reTection) as glancing incidence (for N ¿ 1) is approached. At glancing incidence these rays are of course totally reTected within the approximation of geometrical optics, but these are precisely the limiting rays responsible for the excitation of surface waves. The consequence of high reTectivity inevitably implies slow convergence of these surface-wave contributions. In [5], Nussenzveig examined in detail the Arst two terms of the Debye expansion (and in [6] the third term and e9ects associated with higher-order terms were discussed in connection with both the optical rainbow and glory). The Arst two terms are associated respectively with direct
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reTection from the surface of the scatterer and direct transmission (with no internal reTection), consistent with the physical interpretation of the Debye series. The asymptotic expansions were carried out up to (but not including) correction terms of order (ka)−2 . The behavior of the Arst term is similar to that found for the impenetrable sphere problem, exhibiting (for N ¿ 1) a forward di9raction peak, a geometrical reTection (or lit) region, and a transition region in which the scattering amplitude is expressible in terms of generalizations of Fock functions (see Eq. (4:67) in [5] and Appendix 2), the latter being deAned in [33] as
∞
∞ ei1x e−i1x F(1) = d x + dx : (5.40) Ai(xei =3 ) Ai(xe−i =3 ) 0 0 The second term gives rise to a lit region, a shadow region and again, a Fock-type transition region. By a process termed critical refraction, surface waves are able to take “shortcuts” across the sphere. The application of the modiAed Watson transform to the third term in the Debye expansion of the scattering amplitude is examined by Nussenzveig in [6], along with the e9ect of higher-order terms. It is this term which is associated with the phenomena of the rainbow and the glory. The theory for the Arst two terms is uniformly valid for all N ¿ 1; this is not the case for the third term however. It transpires that there are Ave di9erent ranges of the refractive index to be considered: each of them requires a separate treatment [6,34]. These subdivisions can be found directly from geometrical optics considerations; corresponding to di9erent values of N there are di9erent angular regions characterized by the number of rays—0; 1; 2 or 3—in √ a particular direction within that region. The only range of signiAcance here is 1 ¡ N ¡ 2; the refractive index of water for the optical spectrum falls within this range, and from now on all statements made pertain to it. From a consideration of geometrical optics, there are three di9erent angular regions to examine: (i) a 0-ray (shadow) region located near the forward region, 0 6 ¡ R ; (ii) a 1-ray region near the backward direction, L ¡ 6 ; where as shown below, L = 4 arccos(1=N ) and (iii) a 2-ray region between them, R ¡ ¡ L : However, the presence of (non-geometrical optics) transition regions leads to a total of six regions to examine for ∈ (0; ); the three additional regions arise because of di9raction e9ects at the boundaries between the above three regions. The transition region near = corresponds to the “glory” region, and the 2-ray=0-ray (light=shadow or caustic) transition region corresponds to the “rainbow” region. There is a penumbra region around the 1-ray=2-ray boundary. The so-called “normal” regions can be described as follows. The 1-ray region contains the single geometrical optics contribution plus surface waves excited at the 2-ray=1-ray shadow boundary; these are associated with di9racted rays that take two “shortcuts” across the scattering sphere. This 2-ray=1-ray transition region corresponds to one of the normal “Fock-type” regions. The 2-ray region contains two real saddle points (one for each ray direction in this region) which become complex in the 0-ray (shadow) region; here the scattering amplitude cannot be reduced to a pure residue series. The rainbow light=shadow transition region is associated physically with the conTuence of a pair of geometrical rays and their transformation into “complex rays”; mathematically this corresponds to a pair of real saddle points merging into a complex saddle point. Then the problem is to And the asymptotic expansion of an integral having two saddle points that move towards or away from each other. The generalization of the standard saddle-point technique to
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include such problems was made by Chester et al. [72] and using their method, Nussenzveig was able to And a uniform asymptotic expansion of the scattering amplitude which was valid throughout the rainbow region, and which matched smoothly onto results for neighboring regions (see Appendix D). The lowest order approximation in this expansion turns out to be the celebrated Airy approximation which, despite several attempts to improve upon it, was the best approximate treatment prior to the analyses of Nussenzveig and coworkers. However, Airy’s theory had a limited range of applicability as a result of its underlying assumptions; by contrast the uniform expansion is valid over a much larger range. This new approach also enables a quantitative theory of the glory—a strong enhancement of scattering in the near-backward direction, as noted in Section 3—to be developed. The intensity predicted by geometrical optics is not nearly suYcient to account for this e9ect. In 1947, Van de Hulst conjectured that the glory is due to surface waves that make two shortcuts across the sphere ([50], see also [7]), but no quantitative analysis had been provided. Nussenzveig used the modiAed Watson transform to advantage in the backward scattering direction, and evaluated the residue-series contributions there, showing that they are indeed capable of accounting for the enhancement in the backward intensity, thus conArming Van de Hulst’s conjecture (as far as the third term of the Debye series is concerned; but many other terms contribute to the glory, as summarized in Section 5.7 below). The enhancement arises from the focusing of di9racted rays on the axis which compensates for the exponential damping over the surface of the sphere, and illustrates the phenomenon of “Regge-pole dominance” of the scattering amplitude. Nevertheless, the “third term” theory, highly successful as it is, is not able to satisfactorily account for the backward-scattered intensity as a function of ; it transpires that contributions from higher-order terms in the Debye expansion are necessary to explain the results obtained by numerical summation of the partial-wave series (and note that the fourth term in the Debye expansion corresponds to the secondary rainbow). As noted in Section 3, Bryant and Cox [54] found a very complicated Ane structure consisting of a quasiperiodic pattern with prominent but irregular peaks, and it is this type of behavior that requires higher-order terms to be invoked. Despite some mathematically technical diYculties, it was shown that the quasiperiodic Ane structure could be accounted for in this way, and also resonance e9ects predicted. Furthermore, the same e9ects can be shown to be responsible for the “ripple” in the total cross section for N ¿ 1 (see [10,88,89] and Section 5.6 for more details). Explicitly, the third term in the Debye expansion is, from (5.35),
∞ ∞ i m f2 ( ; ) = − (−1) U ( ; )G( ; )P −1=2 (cos ) exp(2im ) d : (5.41)
m=−∞ 0 After some rearrangement, and a shift in the path of integration to the above real axis, this can be written as
∞+ij i U ( ; )G( ; )P −1=2 (cos ) exp(i ) d ; (5.42) f2 ( ; ) = − 2 −∞−ij cos where H ¿ 0: Furthermore, this integral can be decomposed into f2 ( ; ) = f2;+0 + f2; r = f2;−0 − f2; r ;
(5.43)
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where f2;±0 ( ; ) =
±
i
and 1 f2; r ( ; ) = 2
∞∓ij
−∞±ij
∞+ij
−∞−ij
289
(2) GUQ −1=2 (cos ) d
(5.44)
d : cos
(5.45)
GU exp(2i )
(2) In expression (5.44), Q −1=2 is a Legendre function of the second kind. Details of these manipulations can be found in [33,6]. It can also be shown from the nature of the integrand in (5.44) that there is always some neighborhood of the imaginary axis where that integrand diverges to inAnity. Since this is true for any value of , it follows that there is no domain of -values in which f2;±0 (and hence f2 ( ; )) can be reduced to a pure residue series.
5.4. Geometrical optics rBegimes As previously noted, the third term of the Debye expansion corresponds to rays that are transmitted out of the transparent sphere after one internal reTection. There are in fact three possible types of ray trajectories depending on the value of the refractive index √ N . The one of concern here is for N in the domain (1; 2); in fact it is the subdomain (1; 2) that will be examined below. The other domains are of course (0; 1) and (2; ∞); again, further subdivisions occur, so that there are in total ?ve subdomains: √ √ √ √ (0; 1); (1; 2); ( 2; [6 3 − 8]1=2 ); ([6 3 − 8]1=2 ; 2) and (2; ∞) : The mathematics based on ray theory has been discussed earlier so we merely summarize it here. In terms of the angles of incidence, refraction and deTection (or scattering), i ; r and (= i; r; and Dk , respectively, in Section 1.3) sin i = N sin r ;
0 6 i ; r 6 =2; 0 6 6
and = 2(i − 2r ) + :
(5.46)
For the deTection angle to be no greater than radians it is necessary that , = i − 2r 6 0; the limiting angle of incidence ia such that , = 0 satisAes the implicit condition sin ia a (5.47) i = 2 arcsin N or explicitly cos ia = 12 (N 2 − 2)
(5.48)
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Fig. 15. A schematic representation of the six angular regions associated with both rainbow and glory scattering from a spherical water droplet. The glory, rainbow and 2-ray=1-ray transition region are greatly√exaggerated for clarity. The angular size of these regions are noted (see text for details). This is valid for 1 ¡ N ¡ 2: The rainbow angle is R and L is the 2-ray=1-ray boundary angle. (Redrawn from [6].)
(excluding the zero impact parameter case cos ia = 1). Since i ∈ [0; =2] it follows that such an √ angle ia only exists for N ∈ [ 2; 2]: In terms of ,, it can be seen that d, 2 sin i =N 2 ¿0 ; = dN 1 − (sin i =N )2
√ i.e. , is an increasing function of N . Since , √= 0 for N = 2, it follows that the range of interest here corresponds to , ¡ 0, i.e. N ∈ (1; 2): Note also that the limiting incident ray is
1 i = ; r = arcsin = l ; 2 N the corresponding scattering angle is L = 2 − 4 arcsin
1 1 = 4 arccos : N N
(5.49)
(5.50)
Finally, as noted in Section 1.3, the rainbow angle (the minimum value of the scattering angle) is, in the present notation, sin i; R R = + 2i; R − 4 arcsin N where
1=2 2 N −1 : i; R = arccos
3
The angle L is a 1-ray=2-ray shadow boundary; the rainbow appears around the angle R , which is a 2-ray=0-ray shadow boundary (see Fig. 15).
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5.5. Saddle points The location of the saddle points is a continuing theme throughout the papers [5,6,33] and subsequent work from the author (summarized in [10]); there are many details and subtleties that cannot be entered into in a review of this nature. Nevertheless, the basic features of the process can be described, because there are similarities between the methods employed for the impenetrable sphere problem and for each of the Arst three terms in the Debye expansion (and in principle for the higher-order terms). The method of steepest descent (or saddle-point method) is described in many texts on asymptotic methods (e.g. [103,104]; a brief but excellent summary may be found in [23]) so it will suYce here to note that the integrals of interest are all reducible to the following generic form (neglecting multiplicative factors)
C(w1 ; w2 ; ; ) exp[i !(w1 ; w2 ; )] d w1 ; (5.51) where = sin w1 and = 5 sin w2 : The derivation of the speciAc forms for C(w1 ; w2 ; ; ) and !(w1 ; w2 ; ) involve the Debye asymptotic expansions for H (1; 2) ( ) [33]; these expansions are valid in an oblong-shaped region in the -plane (see [105] for an explanation); this means that the method applied to integrals of type (5.51) above can locate both real and complex saddle points (if any exist) in this region. It transpires that 9! 1 = 2 cos w1 2w2 − w1 − ( − ) ; (5.52) 9w1 2 which yields the saddle points w1 = i ; w2 = r (on using the deTection angle formula (5.46)), and hence = sin i = 5 sin r :
(5.53)
In an appendix to [33], Nussenzveig presents the details calculating the real and complex roots of (5.52) and (5.53); this involves some very complicated algebraic manipulation so a summary will suYce here. For ∈ (L ; ); the 1-ray region in Fig. 15, there is only saddle point, and this is real. As decreases from in this interval the saddle point moves from the origin along the positive real axis to the point zL : At = L another saddle point appears at = = 1, and as continues to decrease from L to R , the two saddle points move toward each other (this is something of an oversimpliAcation; refer to [6], Section 3 for details). They merge at the rainbow angle R ; so for in the interval (R ; L ); there are two real saddle points, corresponding to the 2-ray region in Fig. 15. The remaining interval is ∈ (0; R ): Here the two saddle points become complex, leaving the real axis orthogonally, and tracing complex-conjugate trajectories (see Fig. 16). This is associated with the 0-ray region in Fig. 15. Note that these results are valid only for the domain illustrated in Fig. 15; the exterior of this domain corresponds to the geometrical shadow region; a di9erent representation of integral (5.51) would be necessary to trace the behavior of the saddle points in this region. In fact, in this region the contribution to the scattering amplitude from the complex saddle points no longer dominates the other contributions, so such analysis is not necessary.
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Fig. 16. The coalescence of two real saddle points in the complex -plane as the rainbow angle R is approached from below (i.e. from the lit side). At R the points collide and subsequently move away from each other along complex-conjugate directions as increases away from R into the shadow region. It is the lower complex saddle point that contributes to the wave Aeld in this region.
It has been noted already that there is √a total of six angular regions in (0; ) of interest in the full scattering problem for 1 ¡ N ¡ 2; the regions [0; R ); (R ; L ) and (L ; ] discussed above, together with their transition regions and a transition region near the backward direction (the glory region). We defer discussion of this and the rainbow region and brieTy discuss the nature of the remaining four so-called “normal” regions. 5.5.1. The 1-ray region This is the region − L 6 = (2= )1=3 ;
− −1=2 ;
(2) where the latter restriction is to enable the asymptotic expansion for Q −1=2 (cos ) to be utilized in (5.44), leading to the integral representation (5.51). The path of steepest descent crosses the real axis at the (unique) saddle point at an angle of =4, as it passes from the third quadrant through the fourth to the Arst quadrant. The third term in the Debye expansion (referred to here as f2 for brevity) is dominated by the WKB approximation, representing the contribution from one geometrical ray (transmitted after one internal reTection; see Fig. 1(a)).
5.5.2. The 2-ray region This is the region L − 6;
− R 62 =M ;
where the second restriction is related to the condition for validity of the Debye asymptotic expansion. Essentially, it requires that the “range” of the (now two) saddle points [103] be much smaller than their separation distance (see Appendix D). In this region the path of steepest
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293
Fig. 17. An incident ray AB is reTected in the direction to P; this provides a physical interpretation of the left saddle point associated with equation ` = cos(=2); b is the impact parameter of the ray AB reTected to P. (See [33,34].)
descent must cross the new saddle point at an angle of − =4, so a new path of integration is necessary. Again, f2 is dominated by the WKB approximation, which now represents contributions from the two saddle points; physically, this corresponds to the existence of two di9erent impact parameters giving two parallel geometric rays ([34]; see also Fig. 17). 5.5.3. The 0-ray region This is the region R − 62 =M and here the saddle points become complex. In view of this, the path of integration must now be taken over the one which induces an exponential decreasing contribution in the shadow region; this turns out to be the lower saddle point. This may be interpreted as a “complex ray” [106]. In this shadow region, f2 cannot be reduced to a pure residue series because of the complex saddle points. Towards the lower end of the above domain, the amplitude is dominated by the background integral, which in turn is dominated by the lower saddle point (see Fig. 16). Thus “complex rays” are born, corresponding to super-exponential damping in the shadow (but with a di9erent factor from those associated with the poles n ) [39]. The damping factor is proportional to exp[ − c(| − R |=62 )3=2 ] ;
where 6 is a positive constant. As one moves more deeply into the shadow from the penumbra region the residue contribution, being more weakly damped, eventually dominates the complex-ray contribution.
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Fig. 18. (a) The two types of vertices associated with an internally incident ray at the critical angle l = arcsin(1=N ): In (i) the surface interaction is just total internal reTection; in (ii) it is critical refraction to the outside, followed by a surface wave arc of angle , (which may take any value) and then critical refraction back into the sphere. Combinations of these two types of vertices provide several classes of “diagrams” with p − 1 vertices that contribute to a Debye term of order p. (Redrawn from [47].) (b) Di9racted rays T2 T2 P excited at the shadow boundary T2 B. (Redrawn from [34].)
5.5.4. The 1-ray=2-ray transition region This is the region | − L | . 6 ;
which is a normal (Fock-type) transition between the 1-ray and 2-ray regions. At such a shadow boundary, surface waves are excited (see Fig. 18). A tangentially incident ray undergoes critical refraction and one subsequent internal reTection before emerging tangentially to deAne the shadow boundary. The surface waves excited at this point of emergence correspond to residues at the poles n and propagate into the shadow (1-ray) region. Nussenzveig points out that there are two types of such di9racted rays, corresponding to one or two surface-wave arcs (see Fig. 18) before reemergence as a surface wave. Both types of wave must of course be accounted for in any contribution summations [34]. Now we are in a position to discuss the two remaining and extremely physically signiAcant regions: 5.5.5. The rainbow region (the 2-ray=0-ray transition region) This is the region | − R | . 62 =M :
The mathematical problem here involves the asymptotic evaluation of an integral arising in the 2-ray region, but in a domain where its two saddle points are very close to each other. The
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Fig. 19. The deformed contour of integration (solid line) in Eq. (5.44) for the 2-ray region, i.e. R ¡ ¡ L . Plain crosses correspond to the poles of GU , crossed circles to poles of Q (2) −1=2 and dots in circles to saddle points. The original contour is shown as a broken line; it is deformed into the steepest-descent path, giving rise to residues at the poles − n and n : See [6], Section 4 for details of the corresponding analysis for the rainbow region.
integral is the “geometrical optics” term
i /2 ∞ (2) f2; g ( ; ) = − GUQ −1=2 (cos ) d ;
−/1 ∞
(5.54)
where the path of integration in (5.44) from (−∞ − iH; ∞ + iH) is deformed into the path of steepest descent (Fig. 19) denoted by (−/1 ∞; /2 ∞): (This entails moving across the poles − n and n so the residue contributions also di9er from those in the 1-ray region.) The relevant theory for this asymptotic evaluation (brieTy reviewed in Appendix D) has been developed by Chester et al. [72]. The above integral may be rewritten as 2 1=2 −i =4 f2; g ( ; ) = 2e N F( ; ) ; (5.55) sin where
/ ∞ 2 F( ; ) = g(w1 ) exp[2 f(w1 ; )] d w1 (5.56) −/1 ∞
and
− sin w1 f(w1 ; ) = i 2N cos w2 − cos w1 + 2w2 − w1 − 2
and 1=2
2
g(w1 ) = (sin w1 ) cos w1 cos w2
cos w1 − N cos w2 [1 + O( −1 )] : (cos w1 + N cos w2 )3
(5.57)
(5.58)
The limits −/1 ∞ and /2 ∞ are the images of −/1 ∞ and /2 ∞ in the w1 -plane, respectively. The two saddle points are given by w1 = i and i : In the present domain, = R + H;
|H|1 :
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If we take H ¿ 0 initially, then the saddle points are real. To illustrate the complexity of the calculations, note that after a great deal of analysis it can be shown that [6] 1=2 f2; g ( ; ) = 4ei =4 N (2 )1=6 exp[2 A(j)]{p0 (j)Ai[(2 )2=3 >(j)] ; sin q0 (j) − Ai [(2 )2=3 >(j)]}[1 + O( −1 )] ; (5.59) (2 )1=3 where A(H) = i[N (cos r + cos r ) − 12 (cos i + cos i )] ;
(5.60)
3=2 2 = i[N (cos r 3 [>(H)]
(5.61)
p0 ; q0 =
e−i =4
− cos r ) − 12 (cos i − cos i )] ;
sin i 2 cos i − N cos r
1=2
(cos i − N cos r ) 4N (cos i + N cos r )3 1=2 − N cos ) sin i ( cos r i ± (2N cos i cos r )3=2 [1 + O( −1 )] N cos r − 2 cos i (cos i + N cos r )3 (5.62) >
±1=4
(2N cos i cos r )3=2
and the ± signs correspond to p0 and q0 ; respectively. Also recall that 1 and 1 are the saddle points. The expressions for A(j) and >(j) correspond to half the sum and half the di9erence of the optical paths through the sphere, respectively. Very close to the rainbow angle, i.e. for |j|1, the approximations for the above quantities may be substituted into the expression for f2; g ( ; ) to yield a still very complicated expression (see [6] for details); the result is a good approximation for |j|6, containing as it does, the rainbow region. The dominant contribution to the third Debye term for the scattering amplitude Sj2 ( ; ) turns out to be proportional to
7=6 : The above approximation can still be employed over part of the 2-ray region, where the Airy functions may be replaced by their asymptotic expansions for negative arguments, which we will not state here. The resulting expression is oscillatory (containing a sinusoidal term) and is associated with the interference between the two geometrical ray contributions, giving rise to the supernumerary bows referred to in the introduction. Other expressions, valid for still larger scattering angles in the 2-ray region, may be derived which coincide exactly with the results obtained by the saddle-point method, so the Chester–Friedman–Ursell method leads to a uniform asymptotic expansion, matching smoothly the result obtained by the saddle-point method in its domain of validity. In the 0-ray region, within the domain 62 =M |j| −1=2 , j ¡ 0 so the asymptotic expansions for Airy functions with positive arguments may now be used, resulting in an expression for the complex-ray contribution (described above) and giving rise to Alexander’s dark band between the primary and secondary bows. Further away from the rainbow angle, the residue-series contributions dominate the amplitude in the deep shadow region.
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Airy’s approximation may be recovered from the |j|1 version of Eq. (5.59) for f2; g ( ; ), as expected; this is 16 (3)7=12 −i =4 N 2 (N 2 − 1)1=4 e
1=6 27 (2)1=3 (8 + N 2 )1=2 (4 − N 2 )2=3 i
2 1=2 2 1=2 ×exp √ [6(N − 1) + (4 − N ) ( − R )] 3 (N 2 − 1)1=2 2 2=3 ( − R ) : ×Ai − (4 − N 2 )1=6 3
f2;(Airy) ( ; ) = − g
(5.63)
Van de Hulst [7] claims that this approximation is a useful quantitative theory only for ◦
¿ 5000 (or drops of radius greater than about 0:5 mm) and |j| ¡ 0:5 ; although Huygen’s principle may still be applied for ¿ 2000, “a quantitative theory of the rainbow for the entire gap 30 ¡ ¡ 2000 is still lacking”. The above contributions from Nussenzveig and coworkers bridge this gap and more; indeed, to quote from [10, Section 8:6], the CAM theory bridges the gap between short-wavelength and long-wavelength scattering, remaining applicable all the way down to ∼ 1 : : : For ¿ 100, the accuracy of the CAM approximation is better than 1 ppm, i.e. it becomes, for all practical purposes, more accurate than the ‘exact’ partial-wave expansion! (For further comments on accuracy see Section 5.7 below). For of the order of a few hundred, within the domain |j|6, the corrections to Airy’s theory can attain several percent; this increases with deviation from the rainbow angle. 5.6. The glory The post-Airy study of both the rainbow and the glory may be divided into two phases, expressed succinctly as “. CAM” and “& CAM”. The dividing line for the purposes of this review occurs around the late 1970s=early 1980s, and although what follows below pertains more to the Arst phase, it contains within it much that is relevant to the second phase, since during the latter many of the implications of the CAM method for rainbows and glories were “Ane-tuned” to a high degree. The details that follow are based predominantly on the analysis in [6], which provides a convenient overview of the results in the Arst phase of study. The ‘second phase’ is summarized in Section 5:7 below. The angular region where the glory is observed is = − j; 0 6 j . −1=2 ; i.e. in the neighborhood of the backward region. It is a strong enhancement in this direction due to very small water droplets, with value up to a few hundred ( ∼ 160 corresponds to water droplets with average diameter ≈ 0:028 mm). The e9ect is concentrated within a very small solid angle ◦ around the 180 direction, corresponding to a narrow peak in the back-scattered intensity. This explains in part the unusual meteorological manifestations that have been observed (see the introduction for more details). It is not observed for larger water droplets ( ∼ 1000). Other important features of the phenomenon are as follows:
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If Hi denotes the angular radius of the ith dark ring (as observed in yellow light) then the approximate bounds arise [6,7]: 0:35 . j1 = j2 . 0:45;
1:6 . j3 = j2 . 1:7
and j1 = O( −1 ), which is a measure of the narrowness of the backward peak. The above ratio bounds are di9erent from those associated with another di9raction e9ect: di9raction coronae, which correspond to the forward di9raction peak. The intensity of the rings in the glory decreases more slowly away from the center than it does in di9raction coronae. The radii and brightness of the rings may change with time, even during the course of a single observation. There are indications that the glory, like the rainbow, is strongly polarized. To recapitulate some points made earlier in Section 3, Bryant and Cox [54] made some ◦ numerical calculations of the intensity at or near 180 for the electromagnetic problem; the intensity was computed as a function of ; at intervals of 0:005; near = 200 and 500: The results showed a great deal of Ane structure; in particular they found a rapidly varying quasiperiodic pattern superimposed on a more slowly varying background for the back-scattered intensity. The period for these Tuctuations was found to be in the approximate range 0:81 . . 0:82: Even within a single period, they found irregular peak widths ranging from ∼0:01 to ∼0:1; with intensities changing by a factor of ∼100, resulting in large spikes superimposed on the background. They also found that the total cross section showed Tuctuations similar to those of the ◦ background intensity, but greatly reduced in magnitude; at 90 intermediate-sized Tuctuations were noticed but with periods twice those given above for . The scalar theory developed by Nussenzveig [5,6] was in principle capable of explaining the above features, with the exception of polarization e9ects; however, the details of the intensity variations require that higher-order terms in the Debye expansion be taken into consideration. Van de Hulst [50] conjectured that surface waves are responsible for the glory: speciAcally di9racted rays excited at the 2-ray=1-ray shadow boundary which travel along a surface arc of ◦ about 14 (for water) before emerging in the backward direction, having taken two shortcuts across the sphere (see Fig. 18(c)). Using the formalism of the modiAed Watson transform f2; g ( ; ) may be written as [6]
i /1 ∞ f2; g ( ; ) = GUP −1=2 (−cos )ei tan( ) d ; (5.64)
0 upon employing symmetry arguments. This background integral represents near-central rays which are reTected back after one internal transmission; the surface waves are “generated” by the residue-series contributions f2; res at the poles n : In [34] it is pointed out that the evaluation of f2; res in the -domain under consideration is diYcult because it involves computing a residue at a triple pole of Hankel functions with complex index in a region where several terms in their ◦ asymptotic expansion may have to be retained. Nevertheless, for = 130 and = 180 , f2; res (130; ) ≈ −0:165 + 0:483i : The geometric ray contribution to this order of approximation consists of two terms: the contribution from the directly reTected ray (f0; g ) and that from the central ray which undergoes
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one internal reTection (f2; g ): Thus (for N = 1:33) fg = f0; g (130; ) + f2; g (130; ) ≈ 0:101 + 0:176i : The “exact” scattering amplitude can be computed from its partial-wave expansion with the result that f(130; ) ≈ −0:011 + 0:798i : Noting that |fg |2 ≈ 0:04 while |f|2 ≈ 0:64; it is clear that the geometrical optics or lowest WKB approximation accounts for only about 7% of the total intensity and has the wrong phase. The remainder must be accounted for by the residue-series contributions; this is what has been referred to above as strong “Regge-pole dominance” of the scattering amplitude (but in contrast to earlier circumstances, this is a lit region as opposed to a shadow region where the amplitude would vanish in the WKB approximation). Indeed, the above f2; res term alone corrects the phase in the right direction, and |fg + f2; res |2 = | − 0:064 + 0:659i|2 ≈ 0:44 ;
which is about 70% of the correct value. This implies that the Van de Hulst conjecture is substantially correct. The remaining di9erence must come from residue-series contributions of higher-order terms in the Debye expansion. These are not damped as rapidly as the geometrical optics terms, due to the high internal reTection coeYcient. The angular distribution for these contributions near the backward direction follows from the approximation [107] P n −1=2 (−cos ) ≈ J0 ( [ − ]) = J0 ( j) ;
(5.65)
where, as noted earlier, n ≈ + e i=3 6−1 x n and J0 is a Bessel function of the Arst kind of order zero. This of course corresponds to an intensity distribution near the backward direction proportional to J02 ( j), which is not unreasonable when compared with the observations [7]. In fact, this will explain the slow intensity decrease at large angles, in contrast with di9raction coronae, for which the intensity is proportional to J12 ( )=( )2 : In the former case, the intensity decreases like ( j)−1 ; whereas in the latter case it decreases like ( )−3 : Furthermore, the ratios of the dark-ring radii in the glory are given by the ratios of the corresponding zeros of J0 (x); namely j1 = j2 ≈ 0:44;
j3 = j2 ≈ 1:6 ;
which compares quite favorably with the observational results quoted earlier from [7]. There are two other pieces of evidence which tend to support the general correctness of the Van de Hulst surface-wave conjecture, the numerical evidence being that when Bryant and Cox [54] examined the partial sums of the Mie series as a function of the number of partial waves retained, they found in the near-backward direction that the most important contribution by far comes from the edge domain, l− . l . l+ , which corresponds to nearly grazing incident rays. The other piece of evidence is experimental; Fahlen and Bryant [55] observed a luminosity of the circumference of a water droplet as seen from the backward direction, analogous to the
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luminosity of a di9racting edge as seen from the shadow. This provides direct evidence for the ◦ existence of intense surface-wave contributions along the 180 direction. With the surface-wave explanation in principle fairly well established, we can now ask why ◦ the exponential damping of these waves along a 14 arc would not prevent them from making such a signiAcant contribution. It turns out that, while the speciAc surface waves referred to by Van de Hulst are nonetheless signiAcant, such contributions from higher-order terms in the Debye expansion (by now not surprisingly) also play an important role in this phenomenon. We will not delve further into the mathematical details here, merely suYcing ourselves with some qualitative comments. It can be shown that the total residue-series contribution has the same order of magnitude as the second term in the Debye expansion [6]. Certain “resonance” phenomena may occur among di9racted rays, wherein each di9racted ray returns to√ its starting point after an integer number of shortcuts taken through the sphere. Thus for N = 2; (outside the scope of interest in this article) a resonance occurs after four shortcuts, i.e. the internal piecewise path is a square. For N = [cos( =5)]−1 ≈ 1:236; the corresponding resonance path is a regular pentagon. The character of oscillations in the scattering amplitude can be explained theoretically, in part at least, by the deviation (or otherwise) of the refractive index from one or other of these resonance values. In fact the intensity is very sensitive to the values of and N; which may also explain the variability of the glory referred to earlier (see further comments on this below). The magnitude of the residue-series-dominated resonance peaks predicted by the theory is bounded by |fres ( ; )| . 1=3 ;
which can be appreciated by considering the approximation to the partial-wave series l=l+ 1 1 l fres ( ; ) ≈ (−1) l + [Sl (k) − 1] : i
2
(5.66)
l=l−
It follows from the unitarity condition for the S-matrix that |Sl (k) − 1| 6 2 ;
where the upper bound is attained when the lth partial wave is resonant. Using this and the expression for the edge domain (or edge strip) l− ∼ − c 1=3 . l . l+ ∼ + c 1=3 ;
c = O(1) ;
leads to the above result. Physically, it arises because most of the partial waves within the edge domain are close to resonance (see Fig. 13). We have seen that the di9racted-ray resonances are associated with poles of the Debye expansion; resonances in particular partial waves correspond to Regge poles close to the real axis. Thus it appears that there is an interesting two-way relationship between the Debye and the partial-wave expansions: surface-wave resonances may be interpreted as the cumulative e9ect of many partial waves close to resonance, and vice versa. For the forward scattering amplitude, an irregular Tuctuation known as the “ripple” is known to occur, which Van de Hulst suggested may be a type of “forward” glory [7]. This is reTected in the total cross-section, via the optical theorem. By summing the contributions for = 130
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from the forward di9raction peak (fd ), the Fock correction terms (fF ) and the geometrical-optic component (fg ; neglecting non-central rays), it is found that fd (130; 0) + fF (130; 0) + fg (130; 0) ≈ −2:142 + 69:093i for N = 1:33: The corresponding “exact” result from the partial-wave series is f(130; 0) = − 2:529 + 68:988i ; so that the residue-series contribution must of necessity be fres (130; 0) ≈ −0:387 − 0:105i ; which is in fact smaller than any of the three terms in the approximate sum above. The total cross section /tot is related to f( ; 0) via the optical theorem, i.e. /tot =
4 a2 Im f( ; 0) :
(5.67)
In terms of the above notation we may write, in an obvious manner /tot = /d + /F + /g + /res : The di9raction, Fock and geometric terms give rise to a slowly varying background (which in the absence of the geometric contribution is monotonically decreasing) which approaches /d asymptotically as → ∞: Since fd = i =2 and hence /d = 2 a2 ; this is equivalent to the well-known result that the asymptotic cross-section is twice the geometrical cross section. The geometrical-optic contribution gives a set of relatively slow oscillations with period g = N −1 and amplitude decreasing like −1 , superimposed on the background. Physically, these oscillations arise from interference between waves di9racted around the sphere and those transmitted through it (geometrically). No ripple is to be expected for N ¡ 1 (and none is observed); di9racted rays cannot take any shortcuts through the sphere in this case. However, damping of the ripple is to be expected for an absorbing sphere (complex N; Re N ¿ 1), and such attenuation has indeed been observed in numerical computations [54]. For small values of (here meaning . 4) the ripple peaks may be explained as resonances in successively higher partial waves. As increases however, more than a single partial wave may be near resonance, and the ripple may be thought of more reasonably as a surface-wave phenomenon. In a similar manner to the discussion of the glory, these two pictures are actually symbiotic in that each e9ect in one description corresponds to a collective e9ect produced by several terms in the other one. There is no reason why ripple e9ects should be conAned to just the backward and forward directions, of course. As pointed out ◦ earlier, they have been observed in numerical calculations [54] for = 90 with corresponding ◦ ◦ period twice that found for = 0 and 180 : The amplitude of the ripple contribution has been estimated [6]; for low values of and N = 1:33 the dominant contribution arises from the
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di9racted rays taking four shortcuts (the “square” resonance mentioned earlier). Under these circumstances the contribution to the scattering cross section from the pole 1 is /res ∼ A( ) −2=3 exp(−>4 Im 1 ) sin(4 Re + 4) ;
(5.68)
where the various terms are as follows: A is a bounded amplitude factor, >p is the minimum angle described by surface waves after p shortcuts through the sphere before emerging in the direction ; = (N; ; 1 ; >2 ) is a complex number describing the period of the ripple oscillation, and 4 is a constant phase term. From this expression it follows that the relative amplitude of ◦ ◦ the ripple component is at most of order −2=3 at = 0 , and at most of order 1=3 at = 180 (see [6]) where the ripple is dominant. In directions far from these two angles Nussenzveig concludes that the relative amplitude of the ripple is of order −1=6 , i.e. the geometric mean of the values in the backward and forward directions. This is in good agreement with numerical estimates made by Penndorf [41], who found that the ripple is present in all directions, with ◦ ◦ increasing relative amplitude as increases from 0 to 180 : He estimated the average relative ◦ ◦ ◦ amplitude as 0:1; 5 and 500, respectively, at = 0 ; 90 and 180 : 5.7. Summary of the CAM theory for rainbows and glories The modiAed Watson transformation is a very e9ective tool in the high-frequency domain ( 1=3 1; 1=3 |N − 1|1=2 1) for extracting the complete asymptotic behavior of the scattering amplitude in any direction from the partial-wave expansion. In general, the total amplitude in any direction is found by adding the contributions from the Arst three terms in the Debye expansion and incorporating, where appropriate, the higher-order correction terms. There are six subdivisions of the interval (0; ) that arise in the rainbow=glory problem (i.e. corresponding to the third term in the Debye expansion) for N ¿ 1. Essentially the regions consist of those predicted by geometrical optics and transition regions; the dominant term in the lit regions is usually that provided by the geometrical-optic contribution, which is the Arst-order WKB approximation. (The Arst correction term is the second-order WKB approximation.) In the shadow regions the surface-wave contributions usually dominate, and this is usually pictured in terms of geometrical di9raction. Surface waves are excited by tangentially incident rays (for N ¿ 1) for both the impenetrable and transparent sphere problems, though in the latter case, as noted earlier, there are two types of surface interactions from within the sphere wherein shortcuts may occur across the sphere. There are three types of transition problem of interest here: (i) normal transitions (of the generalized Fock type); (ii) the rainbow; (iii) the glory. Type (i) transitions also arise for the impenetrable sphere, and in both cases their angular width is of order 6: They include the region around the forward di9raction peak. The rainbow region (ii) is associated with the transformation of a pair of real rays into complex rays, and a uniform asymptotic expansion can be developed for the amplitude which contains the Airy theory as a special case, but which is valid for a considerable domain beyond that approximation. Finally, the glory region (iii) is an example of “Regge-pole dominance” of the scattering amplitude in near-backward directions. The glory is due to surface waves taking shortcuts through the sphere, but with some corrections (from higher-order terms) to the original idea proposed by Van de Hulst. With the exception of polarization, the CAM version of the theory enables an almost complete description
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of the glory to be made. The features so explained arise from competition between four di9erent phenomena: (a) exponential damping of surface waves as they travel along the spherical surface; (b) focusing of di9racted rays along the axis, which enhances the back-scattered contribution; (c) high internal reTectivity of di9racted rays at the surface, which means that many internal reTections must be taken into consideration in general; (iv) resonance e9ects associated with nearly closed circuits after four successive short-cuts. Estimates of higher-order residue-series contributions can also be made. The quasiperiodic intensity Tuctuations associated with these resonant di9racted rays are present in all directions, but dominate near the backward direction in terms of large intensity variations that characterize the glory region. The contribution from surface-wave terms decreases with increasing , eventually (for large enough ) permitting the geometrical-optic terms to become dominant again. The electromagnetic problem is discussed in some detail by Khare and Nussenzveig in [47,108] (see also [53]). Nussenzveig critiqued his (‘phase one’) papers from mathematical, numerical and physical points of view [6] in the sense of suggesting further improvements, extensions and applications, which were later addressed in [10] (and references therein). Nevertheless, several of his earlier comments in this regard bear repeating here: in the neighborhood of transition points between di9erent ranges of N , a conTuence of saddle points may occur near = , leading to a mixture of rainbow and glory e9ects; this has been discussed more fully in [47]. Furthermore only the scattering amplitude has been studied—the behavior of the wavefunction itself has not been discussed (as it has been for the impenetrable sphere problem [33]). The extension of the theory to complex N to represent an absorptive sphere is of interest, both from a mathematical point of view (the convergence of the Debye series would be improved owing to increased damping), and because of potential applications to both nuclear physics (in the optical model) and atomic physics (see Section 6.5). Indeed, there was also early evidence for nuclear glory scattering [109]. Extension of the theory to both di9erent-shaped scattering objects and inhomogeneous bodies (see for example [110]) is of considerable interest (both of which correspond to more general potentials than the square well=barrier in the quantum–mechanical context). The second phase of the analysis of the glory (from about 1980 onward) Alled in the gaps that were present in the early studies. The later work showed that higher-order terms in the Debye expansion, both from surface waves and the shadow of higher-order rainbows, cannot be neglected [10]. Noting that the glory is one of the most complicated of all scattering phenomena [10], Nussenzveig lists (along with their explanation) a total of 15 observational and numerical features associated with optical glories (some of which are conspicuously di9erent from coronae (which arise from a forward di9raction peak); indeed, as already noted, van de Hulst named them anti-coronae [50]). BrieTy summarized, they are: (i) variability of the glory rings; (ii) their angular distribution; (iii) polarization (parallel polarization appears to be dominant in the outer rings); (iv) the angular width of the glory region is O( −1 ); (v) features around = 102 ; (vi) features around = 103 ; (vii) intensity enhancement (at least an order of magnitude greater than that predicted by geometrical optics; this is due to dominant di9raction e9ects); (viii) ripple Tuctuations; (ix) quasiperiodicity; (x) background intensity; (xi) e9ects of absorption or size averaging; (xii) other periodicities; (xiii) quasichaotic features; (xiv) average gain factor (the ratios of polarized intensities to their limiting value as predicted by geometrical optics for a totally reTecting sphere); and (xv) edge origin (dominant contributions to the backscattered intensity arise from the edge strip).
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Within the glory region, cross-polarization e9ects also contribute to the intensity enhancement. Concerning surface waves, van de Hulst’s p = 2 term is an important contributor to the optical glory, but which higher-order Debye terms might contribute signiAcantly to backscattering? It is known that in general such terms are strongly damped by multiple internal reTection except within the edge strip(s)
6 6 (1 + c+ 62 ) and
(1 − c− 62 ) 6 6 ; i.e. near the top of the centrifugal barrier (as noted in Section 5.1 the constants c± are of order unity; note also that ‘physical’ values of correspond to = l + 12 ; in Sections 4 and 5 and [5,6], the edge strip is stated in terms of l; in [10] the above form is used). It transpires that Debye terms of orders up to a few times 1=3 (undergoing a large number of internal reTections in a manner analogous to orbiting) can yield appreciable contributions from these regions. In addition there are enhancements due to axial focusing (of order 1=2 arising from constructive interference along the axis from toroidal wavefronts where the edge strip acts as a virtual ring source) and proximity to an edge backward glory ray or a near-backward higher-order rainbow (of order 1=6 ). The latter arise from near-edge incidence; the 10th-order rainbow is predominant, being formed very close to = , and for &200 this is the leading term. However, the width of the main rainbow peak decreases as increases, and complex-ray damping in the rainbow shadow reduces the backward gain factor [10]. The quasiperiodic features referred to above are consequences of interference oscillations among Debye components (and corresponding shortcuts through the droplets); the quasichaotic features are identiAed with the sensitive dependence of lower-order geometric resonances (for closed or nearly closed orbits) on the droplet size and physical parameters, as are the resonances in the total amplitude (the ripple Tuctuations). The slowest ripple Tuctuations are also major contributors to the glory; they correspond to resonances in the total scattering amplitude (s). Because an inAnite number of branches of the deTection function contibute to this component of the glory (in keeping with the concept of orbiting), they cannot be represented as a Anite sum of terms in the Debye expansion. The resonances arise from complex Regge poles near = (see Fig. 19) as a result of tunneling around the edge strip. Thus (and leading naturally to the next subsection) it has been stated that [10]: ...all leading contributions to the glory arise from complex critical points (poles or saddle points), so that this beautiful and impressive meteorological e9ect is produced almost entirely by light tunneling on a macroscopic scale. 5.8. A synopsis: di;ractive scattering, tunneling e;ects, shape resonances and Regge trajectories [89] For a sphere of radius a and dimensionless size parameter = ka1 (k being the incident wavenumber) the di9ractive scattering amplitude in direction is [65] ia fd (k; ) = J1 ( ) ; (5.69)
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where J1 is the Bessel function of the Arst kind of order one. Nussenzveig [10] notes that the angular width of the forward di9raction peak (and the spacing of secondary peaks) is very narrow, being O( −1 ). The total (elastic) di9ractive cross section [65] is
∞
∞ 2 J1 (z) 2 2 /d ≈ 2 |fd (k; )| d = 2 a d z = a2 ; (5.70) z 0 0 which is the cross sectional area of the sphere. The forward peak arises from axial focusing: secondary waves with di9erent azimuths interfere constructively along the axis. The domain of validity of the above expression is 0 6 6 −1 , and the question therefore arises: how can the domain be extended to larger angles of di9raction? One approach was developed by Fock [112], based on the idea of “transverse di9usion” along a wavefront [113,114]; an alternative approximation is Keller’s geometric theory of di;raction [106,115]. Di9racted rays are introduced as extremals of the optical path, and follow the laws of geometrical optics. A smooth scattering object (e.g. a sphere) can support such rays along segments of the boundary; tangentially incident rays excite such di9racted rays. Considered as surface waves, their amplitudes are damped exponentially with angle of travel () around the surface because of tangential shedding of radiation (in acoustics these waves are sometimes referred to as creeping modes). As noted above, competing approximations can be conveniently tested using the model of scattering (both by direct reTection and di9raction, in this instance) by an impenetrable sphere of radius a (the transparent sphere is not necessary since it is only the “blocking e9ect” that is being described here; obviously the transparent sphere model is crucial for understanding the complete scattering problem). The domain of validity of the semiclassical approximation to the reTection amplitude is determined by the condition that the Arst Fresnel zone for the reTected ray should not intersect the discontinuity at the light=shadow boundary on the surface of the sphere [114]. This imposes the condition 1=3 2 &2 = 26 ;
where 6 is a parameter measuring the angular width of the penumbral region. This is much broader than the classical di9raction domain . −1 (since 1), but both the convergence of the WKB expansion (used in the semi-classical approximation) and the surface wave contributions are rapid only for 6. It is not surprising therefore that the greatest analytical and numerical challenges are associated with the domain between these two angular regions. Fock’s theory yields only a transitional asymptotic approximation here, and is based on an inadequate physical picture (because the di9usion coeYcient is pure imaginary, the di9usion equation should be replaced by the SchrRodinger equation, with corresponding changes in interpretation— see below). A uniform semiclassical approximation for potentials with long-range tails was provided by Berry [116] (though recall that the forward di9raction peak in this case is not caused by edge di9raction), and for Anite-range potentials of the type discussed here, Nussenzveig and co-workers applied complex angular momentum (CAM) theory to obtain the requisite uniform approximation [117–119]. The essential (and novel) feature of his approach is to regard di9raction as a tunneling process.
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The basic procedure used is to transform the partial-wave expansion for the scattering amplitude by means of a modiAed version of the Poincar7e–Watson transform (via the Poisson summation formula [33]). In this process, the scattering function Sl (k) is extended to continuous values of angular momentum in the complex -plane (at this juncture for the impenetrable sphere only; the extension to the transparent sphere was developed later [5,6]). The resulting expression for the scattering amplitude f(k; ) can be considered to be a superposition of “pseudoclassical paths” (generalizations of di9racted rays) [120] associated with all values of (away from the forward and backward scattering directions). The qualiAcation is to distinguish such paths from the corresponding classical paths, the envelope of which is the sphere r = r0 ( ), r0 being the outermost radial turning point (di9racted rays are pseudoclassical paths with r0 ( ) = a; they are composed of incoming and outgoing classical paths). The resulting integral superposition over arises from the classical “action” along such paths (the action being the analog of the optical path). In appropriate units (i.e. ˜ = 2m = 1), the impact parameter associated with angular momentum is b = =k (this follows from the so-called localization principle [7,10]) tangential rays (grazing or edge rays) correspond to the threshold value = with energy k 2 located at the top of the potential barrier in the e9ective potential (see Figs. 29(a), (c) and also Fig. 13). So-called below-edge rays ( ¡ = ka) ‘hit’ the sphere, and above-edge rays ( ¿ ) pass outside with a radial turning point given by r0 ( ) = b( ); it is the contributions from these regions close the edge that give rise to signiAcant modiAcations of classical di9raction. When ¿ tunneling through the ‘triangular’ barrier can occur between r = r0 ( ) and a (see below for further discussion of this); as might be expected, if the potential is approximated by a linear potential, the solutions of the SchrRodinger equation are expressible in terms of Airy functions (more precisely, as generalized Fock functions; see Appendix B). By deforming the path of integration for the scattering amplitude into the complex plane, the integral can be evaluated approximately from a small number of dominant contributions. The outer approximation is valid for 6, being the sum of the WKB expansion (for reTection) and small surface wave corrections. For values of below this range (including zero) the uniform approximation is composed of three dominant terms, namely uniform versions of the classical di9raction amplitude, above-edge amplitude (incorporating the e9ects of tunneling through the centrifugal barrier) and below-edge amplitude (see [121]). The CAM approximation is obtained from a smooth match of the outer and uniform approximations, and is valid for 0 6 6 . It is also extremely accurate [10,121] and bridges the gap between short and long wavelength scattering (remaining accurate down to ∼ 1).
= There is an interesting consequence from the e9ects of above-edge tunneling: each contribution is weighted by the uniform tunneling penetration amplitude through the centrifugal barrier to the surface, and in contrast to classical di9raction, represents a nonlocal interaction with range b − a ≈ ( 02 a)1=3 , i.e. a weighted geometric mean between the wavelength 0 and the radius a. Furthermore, it should come as no surprise by now that near-edge tunneling is also very signiAcant in rainbow and glory scattering and also orbiting. We are reminded in [122] that descriptions of tunneling always involve analytic extensions to complex values of some parameter, and CAM is a natural vehicle for this. In rainbow scattering, for example, penetration into the shadow zone (tunneling) occurs through a complex ray; in the optical glory surface waves (launched by tunneling) and complex rainbow shadow rays make dominant contributions
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to this phenomenon; orbiting also contributes to the glory, and to resonance e9ects, which are well-illustrated by the problem of scattering by a transparent sphere (with real refractive index N ¿ 1). Now the e9ective potential has a discontinuity at r = a, essentially because of the ‘addition’ of a potential well to the centrifugal potential. Thus the ‘spike’ corresponds to a barrier surrounding a well, and suggests the existence of resonances, particularly between the top of the former and bottom of the latter, where there are three turning points. Such resonances are called “shape resonances” (and sometimes these are called “morphology-dependent resonances” [123]); they are quasibound states in the potential well that escape by tunneling through the centrifugal barrier. The widths of these resonances depends on where they are located; the smaller the node-number of the radial wavefunction within the well, the deeper that state lies in the well. This in turn determines the width (and lifetime) of the state, because the tunneling amplitude is “exponentially sensitive” to the barrier height and width [122]. Since the latter decreases rapidly with the depth of the well, the smaller is the barrier transmissivity and the lowest-node resonances become very narrow for large values of . The lifetime of the resonance (determined by the rate of tunneling through the barrier) is inversely proportional to the width of the resonance, so these deep states have the longest lifetimes [123,88] (with theoretical details in [89]). Some calculations are in order at this point. Powerful analogies abound in physics; of particular interest here are the well-known analogies of Hamiltonian mechanics with geometrical optics and wave mechanics with wave optics [10]. In the former case, a non-relativistic particle of energy E (= k 2 ) in a central potential V (r) has associated with it in the optical realm an index of refraction N given by V (r) N = 1− : (5.71) E The analog of the optical path being the action, Fermat’s principle corresponds to the principle of least action. In particular, if the potential is a square well for which V (r) = − V0 ; V0 ¿ 0; for 0 ¡ r ¡ a; then the depth of the well is V0 = (N 2 − 1)k 2 ; this is therefore a potential which is directly proportional to the energy in contrast to the standard quantum mechanical problem in which V (r) is a Axed function, independent of the energy. Thus it follows that if the energy k 2 is varied, so does the depth of the potential well (or barrier, if V0 ¡ 0). The e9ective potential corresponding to the square well is (in units with ˜ = 2m = 1) (3)
2 − (N 2 − 1)k 2 for 0 ¡ r ¡ a r2 2 = 2 for a 6 r ¡ ∞ r
U (r) =
(5.72)
(see Fig. 29). Note that as k 2 is reduced, the bottom of the potential rises (and for some value of k the energy will coincide with the bottom of the well [123]); however, at the top of the well, U (a) = 2 =a2 is independent of k 2 , but if k 2 is increased, it will eventually coincide with the top of the well. Consider a value of k 2 between the top and the bottom of the well: within this range there will be three radial turning points, the middle one obviously occurring at r = a, and
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the largest at r = b for which U (b) = 2 =b2 . The smallest of the three (rmin ) is found by solving the equation 2 k 2 = 2 − (N 2 − 1)k 2 ; rmin to obtain b = (5.73) Nk N in terms of the impact parameter b( ) = =k. By applying Snell’s law for given b, it is readily shown that the distance of nearest approach to the center of the sphere is just rmin ; indeed, there are in general many nearly total internal reTections (because of internal incidence beyond critical) within the sphere between r = b=N and a. This is analogous to orbiting in a ray picture; the very low leakage of these states allows the resonance amplitude and energy to build up signiAcantly during a large resonance lifetime which in turn can lead to non-linear optical e9ects [10]. In acoustics these are called “whispering gallery modes” [124,125]. The energy at the bottom of the well (i.e. limr→a− U (r)) corresponding to the turning point at r = a is determined by the impact parameter inequalities a ¡ b ¡ Na, or in terms of = kb, 2 2 − 2 U (a ) = ¡k ¡ = U (a+ ) ; (5.74) Na a which is the energy range between the top and bottom of the well (and in which the resonances occur). To cross the “forbidden region” a ¡ r ¡ b requires tunneling through the centrifugal barrier and near the resonance energies the usual oscillatory=exponential matching procedures lead to very large ratios of internal to external amplitudes (see Fig. 29(c)); these resonances correspond to “quasibound” states of light (that would be bound in the limit of zero leakage) [88]. Mathematically, the resonances are complex eigenfrequencies associated with the poles n of the scattering function S( ; k) in the Arst quadrant of the complex -plane; these are known as Regge poles (for real ). Corresponding to the energy interval [U (a− ); U (a+ )] the real parts of these poles lie in the interval = ka ¡ Re n ¡ N = Nka (i.e. impact parameters ∈ [a; Na] which is of course the tunneling region). The imaginary parts of the poles are directly related to resonance widths (and therefore lifetimes); in fact Nussenzveig has coined the term “life angle” instead of lifetime, since the leakage gives rise to angular damping [92]. As n decreases, Re n increases and Im n decreases very rapidly (reTecting the exponential behavior of the barrier transmissivity). As increases, the poles n trace out Regge trajectories, and Im n tends exponentially to zero. When Re n passes close to a “physical” value, = l + 1=2, it is associated with a resonance in the lth partial wave; the larger the value of , the sharper the resonance becomes for a given node number n. More speciAc mathematical details are provided below. It has been noted several times that Mie scattering cross sections have a very complicated structure as functions of ; this “ripple” is extremely sensitive to parameter changes in the cross sections (quasichaotic) at all angles, being greatly enhanced in backscattering; but it does contain some quasiperiodic features. It is very much related to the sensitivity of the resonances to variations in barrier shape noted above. A detailed CAM analysis [88,89] identiAes the ripple rmin =
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with the residue contribution from Regge trajectories, and it is to these we now brieTy turn. The poles of the S j ( ; ) function (the scalar version of which is given by Eq. (5.22)) are the roots of the following complex transcendental equation [10] 1 1 (1) (1) ln H ( ) + (5.75) = Nej ln J (5) + 2
25 in the notation of Section 5, where ‘ln ’ refers to the logarithmic derivative, and for the magnetic (electric) polarization, e1 (e2 ) = 1 (N −2 ). For real ; the Regge poles are the zeros of this expression in the complex -plane; for real = l + 12 ; the poles may also be found in the complex -plane, although this is not particularly relevant here. It is the Regge poles close to the real -axis, with ¡ Re ¡ 5, that are of interest (see Fig. 14). The imaginary part of these poles is exponentially small. Their approximate location was obtained in [5,88], and if N is real, their real parts (in the lowest order of approximation) are determined by the implicit relation 1=2
a 2 2 2 2 N k − 2 d r = (2n + 1) (n = 0; 1; 2; : : :) ; (5.76) r =Nk where the integrand plays the role of an e9ective radial wavenumber within the well. The integral is a radial phase integral between the inner turning point and the surface; it is noted in [10] that the above condition is like a Bohr–Sommerfeld quantization condition [20] for ‘bound’ states of electromagnetic energy. The integer n plays the role of a quantum (or ‘family’) number. To the same degree of approximation, the imaginary part of the Regge poles is proportional to the centrifugal barrier penetration factor
1=2 =k 2 2 exp[ − 2M(a; =k)] ≡ exp −2 −k dr ; (5.77) r2 a where M is the radial phase integral between the surface and the outer turning point. Higher-order approximations can be obtained [88]. As varies, each complex Regge pole nj ( ) describes a Regge trajectory (a path in each of the Re = and Im = planes). When Re nj ( ) = l + 1=2 a resonance occurs for polarization j in the lth partial wave. The lifetime of the resonant state is the inverse of its width and proportional to the inverse of the transmissivity of the barrier. As such, the dominant inTuence on resonant widths is that of tunneling through the centrifugal barrier. We revisit non-relativistic quantum scattering of an impenetrable sphere by considering the generalization of the above expression (in connection with the above-edge amplitude). In the notation of Nussenzveig and Wiscombe [118], the uniform approximation to the dimensionless scattering amplitude F( ; ) ≡ f(k; )=a is [117] F = Fd + F˜ s + Fe+ + Fe− ;
(5.78)
where the Arst term is the uniform version of the classical (blocking) di9raction amplitude. The second term F˜ s is the surface-wave contribution from waves that have traversed at least half the circumference, and is usually small in comparison with the last two terms which cumulatively
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are referred to as the edge amplitude Fe ; arising from incident rays with impact parameters in the edge domain. The ‘±’ identiAes the above=below edge amplitudes respectively. Of primary interest here is the former amplitude, Fe+ . By employing uniform asymptotic expansions for both the Legendre function P −1=2 (cos ) and cylindrical functions (see [117] for more details) it is found that
∞ Fe+ ( ; ) = − H(x; ,)P(; cosh ,)(,−1 cosh ,)x1=2 d x ; (5.79) 0
where
P(; ) ≡ H(x; ,) ≡
with
and
sin
1=2
1 −1 J1 ( ) −3 [1 + O( )]J0 ( ) − ( − cot ) + O( ) ; 8 −2
e−i =6 Ai(x) − /(x; ,)Ai (x) Ai(e2i =3 x) + e−i =6 /(x; ,) Ai (e2i =3 x)
5e−i =6 1 3 2 /(x; ,) ≡ − coth , + 5 24 x1=2 sinh , 3(, coth , − 1) 2=3 3
x≡ : (, cosh , − sinh ,) 2
This independent variable is equivalent to 6( − ) ≡ 6k(b − a), where b is the impact parameter associated with angular momentum ; note that |x| . 1 deAnes the edge strip [10]. For x1, H(x; ,) is approximated by
1=2 =k 2 4 3=2 i =3 2 e H(x; ,) ≈ exp − x = exp −2 −k dr : (5.80) 3 r2 a (There is a minor misprint in Eq. (7) of [118].) Hence Fe+ is a pure tunneling amplitude: it describes uniform tunneling of the outside rays through the centrifugal barrier to the surface (with which they interact) and back. The below-edge amplitude Fe− is given by an integral similar to (5.79), and the integrand again has behavior dominated by a nonlinear exponent. For real paths, it corresponds to strong reTection above but close to the top of the centrifugal barrier (containing the e9ects of surface curvature on reTection); but an alternative representation in terms of complex angles [117] demonstrates that this is equivalent to tunneling with the super-exponential decay factor (5.80) (for a brief summary of the contributions from both paths of steepest descent and stationary phase describing the ‘birth’ of a reTected wave in the WKB region, see [10]). Thus, the full edge amplitude Fe can be interpreted as a tunneling amplitude. For . 6; it can be expanded in powers of =6, leading to generalized Fock functions, and the recovery of the Fock approximation. This breaks down, not surprisingly, for 6, and as is characteristic of transitional approximations, cannot be matched with large angle results. However, it can be
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smoothly matched with the outer approximation F = Fr +Fs , where Fr is the reTection amplitude given by the WKB approximation (with corrections up to the second order) and Fs is the surface wave amplitude containing all surface-wave contributions. 6. The electromagnetic problem 6.1. Polarization The scalar theory obviously has nothing to say with regard to polarization. We have noted, however, that the major techniques introduced in the Debye=Nussenzveig scalar approach are applicable also to the electromagnetic problem [59,10]. Much, indeed most of the analysis for the scalar problem carries over to this more general one. The main di9erence, as noted in Section 1.4, is that we now must deal with the Mie solution [24] and the fact that there are now two scattering amplitudes to consider: one for each of the magnetic and electric polarizations. In a change from the f notation for the scattering amplitude, Khare and Nussenzveig [47] use S1 ( ; ) and S2 ( ; ) to denote these, respectively (these forms were used in Section 3.1 also), and we will respect that notation in this section as well. As pointed out in [47,61], the intensities |Sj ( ; )|2 (j = 1; 2) and the phase di9erence = arg S1 −arg S2 completely characterize the scattering. The modiAed Watson transform is applied to each term in the Debye (multiple internal reTection) expansion of the Mie series, i.e. for ∞ (0) (1) (2) Sj = Sj + Sj + Sj + · · · = Sj; p ( ; ); j = 1; 2 ; (6.1) p=0
where Sj; p is associated with waves that undergo p − 1 internal reTections. As in the scalar case, the primary rainbow is associated with the third Debye term Sj(2) , which is associated with rays undergoing a single internal reTection in the spherical drop. Again, the rainbow region is a 2-ray=0-ray transition region, represented in the complex angular momentum plane by the conTuence of two real saddle points then becoming complex; the method of Chester et al. (CFU method: see [72] and Appendix D) once more provides a uniform asymptotic expansion in this situation. The dominant contribution to Sj(2) is given by the same type of expression(s) as stated in Eqs. (5.55) – (5.58) above. The saddle points in the 2-ray region, i and i , correspond to the two angles of incidence associated with geometrical rays emerging from the drop in the direction . The path of integration is the same as in the scalar case. For |(2 )2=3 >|1 (see Eq. (5.59)), the result matches smoothly with those in the neighboring angular regions, in contrast with that from Airy’s theory. Some of the CFU coeYcients are considerably smaller for polarization “2” (parallel; electric) because the angle of incidence for rainbow rays is close to Brewster’s√angle, and this suppresses much of the contribution from that polarization (indeed, for N = 2, the rainbow angle is Brewster’s angle (see [111]: in this paper, which is in German, a variety of rainbow features is discussed, including the refractive indices for which rainbows of all orders are completely polarized, together with brief comments on the ‘appearance’ of rainbows in literature, art, folklore and religion).
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There is of course a contribution to the rainbow region from the ray directly reTected at the drop surface, represented by the Arst Debye term Sj(0) ( ; ), and this was included in the numerical studies undertaken by Khare and Nussenzveig [59]. They compared the exact Mie results (obtained, recall by summing over a number & of partial waves) for N = 1:33; ◦ ◦ 50 6 6 1500, and 136 6 6 142 with both the Airy approximation and the results from system (5.55) – (5.58). The main correction to the Airy theory is provided by the Ai (−x) term (where x = − (2 )2=3 >) in Eq. (5.59). For polarization 1, within the main rainbow peak |x| . 1, the corrections to the Airy theory are small, but for the supernumerary arcs (x1; secondary peaks) they become signiAcant. For polarization 2 there are large corrections to the Airy theory, because the Ai (−x) term is dominant throughout the complete range of interest. Because there is a change in sign of the amplitude reTection coeYcient at Brewster’s angle, one such correction is that the secondary-peak maxima and minima should be interchanged for the two polarizations (see also [7]). The results are compared in several Agures in [59]; in particular ◦ for = 500 oscillations with period ≈ (300= ) are superimposed on the main peak and part of the secondary (see Fig. 2 in [59]); this results from interference with the direct reTection term. Such interference remains appreciable up to = 1500 (the upper value investigated; this is true even close to the rainbow angle R ), so the direct reTection term has been subtracted out in their Fig. 3, which is reproduced here as Fig. 20. For the same value of the phase di9erence is also plotted: the Airy theory fails here, even close to R , whereas the exact results and asymptotic theory agree well throughout. Again, rapid oscillations arise from interference with direct reTection. The authors point out that the oscillatory nature of the deviations between the asymptotic theory and the exact solution is consistent with interpreting them as “ripple”; this, as we have noted above, is present in all directions (but dominant in the glory) is due to grazing incident rays which are in turn associated with Regge-pole-type contributions (surface waves) and higher-order Debye terms. Concerning the glory, the same techniques apply, but for = the scattering amplitudes are related by S1 ( ; ) = S M ( ) + S E ( ) = − S2 ( ; ) ;
(6.2)
where the contributions from the magnetic multipoles (M ) and the electric multipoles (E) are of the same order (this is called the cross-polarization e9ect [7,53,108]). Again, the object is to And which p values in the Debye expansion for each polarization yield signiAcant contributions at a given value of ; and thence to represent these by suitable asymptotic expansions via the modiAed Watson transform. There are in general four sets of contributions from (i) geometrical-optic rays (real isolated saddle-points in the -plane); (ii) surface waves (complex Regge poles); (iii) rainbow terms (conTuent saddle points) and (iv) Fock-type transition region contributions (interpolating smoothly between (i) and (ii)). As we have seen in the scalar case, the contributions from geometrical optics are too insigniAcant to account for the glory; and while van de Hulst conjectured that p = 2 surface waves are the dominant feature in describing the phenomenon, as we have already seen, higher-order Debye terms need to be incorporated for a complete description. For p1, the important contributions come from the edge domain | − | . 1=3 , where the internal reTectivity is close to one. However, there are also contributions from higher-order
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Fig. 20. A comparison of the intensities predicted by the exact Mie theory, the complex angular momentum (CAM) theory and the Airy theory for = 1500, redrawn from [59]. The direct reTection term has been subtracted out. (a) refers to magnetic polarization intensity and (b) to the electric polarization intensity.
rainbows formed near = , due primarily to the 1=6 rainbow enhancement factor [61] (but see comments in [42]) which persists at considerable angular distances from the particular rainbow angle Rp (because the width of the rainbow region increases with p). The backward direction falls within the shadow side of the relevant rainbows; here the amplitude is exponentially damped with exponent proportional to (|>p |p−1 )3=2 for large values of p. The corresponding exponent for the surface-wave terms is 1=3 >p ; note that >p ≡ − pt (mod 2 ) ;
(6.3)
where t = 2 arccos(1=N ), − 6 >p 6 . If >p ¿ 0 this represents the angle described by a surface wave before emerging in the backward direction; if >p ¡ 0 this corresponds to the approximate deviation HRp = − Rp from the rainbow angle. Khare and Nussenzveig [108] conjectured, in view of the damping exponents, that the dominant Debye contributions arise from those values of p for which −t 6 >p ¡ t , and that for surface waves the lowest >p dominate, whereas rainbow terms are dominated by the lowest values of |>p |=p. They carried out numerical comparisons (neglecting the direct-reTection term p = 0) with the exact results for N = [cos (11 =48)]−1 ≈ 1:33007
(6.4)
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Fig. 21. The path of a tangentially incident ray for a refractive index given by N = sec(11 =48) 1:33007 (see Eq. (6.4)). The values of p (corresponding to the pth Debye term) at each vertex are indicated, and the arrows point in the directions corresponding to the surface angles (>p ) traversed. Surface-wave terms are indicated by dotted arrows, rainbow terms by solid arrows. The length of the arrows corresponds qualitatively to the ordering of terms (by increasing >p for surface waves and |>p |=p for rainbow terms).
for which >p = 0, i.e. a tangentially-incident ray becomes resonant by forming a closed 48-sided star-shaped regular polygonal path inside the spherical droplet (see Fig. 21). 6.2. Further developments on polarization: Airy theory revisited In this subsection we examine an important and mathematically simpler approach to this problem formulated by KRonnen and de Boer [126]. It is based on the classical Airy theory but with an important di9erence: a position-dependent amplitude term is introduced for the polarized rainbow (as opposed to the common rainbow—terms to be precisely deAned below). As Airy showed, the wavefront emerging from a raindrop is well approximated by a cubic function near the geometric ray of minimum deviation (the D7escartes ray), and the interference pattern of light from such a wavefront is described by the Airy function Ai(z), where z ˙ R − , i.e. the deviation of the ray from the rainbow angle (Section 2). The Airy approximation is valid provided the deviation angle is only a degree or two, and the raindrop diameter is & 0:3 mm for visible light. As already noted, light from an optical rainbow is strongly polarized; the direction of the E vector is tangential to the rainbow. In the primary rainbow, light is internally reTected once, and for the D7escartes ray this reTection occurs close to the Brewster angle; at this angle light polarized parallel to the plane of incidence is completely transmitted (none reTected) leaving only light polarized perpendicular to this plane in the reTected contribution. This is the tangential component referred to in [126]; the parallel polarization is referred to as radial. Since
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the reTection angle for the D7escartes ray is several degrees away from the Brewster angle, some of the radial component is reTected to produce a correspondingly much weaker rainbow; it is this that we shall refer to as the polarized rainbow, the more intense common rainbow being tangentially polarized. It is interesting to note that the supernumerary bows (intensity maxima) in the polarized rainbow are shifted to the locations of the minima in the common rainbow; this was Arst observed in fog bows by Bricard [127]. From the geometrical optics viewpoint, supernumerary bows are caused by interference between two rays with di9erent impact parameters (and hence path lengths inside the drop) and the same scattering (deviation) angle. If the respective angles of incidence for these two rays, i1 ; i2 say, are such that i1 ¡ IB ¡ i2 , where IB is the Brewster angle, then these two rays have an additional phase dif◦ ference of 180 (see the very useful descriptions in KRonnen’s book [128]). This does not occur for the corresponding rays in the common rainbow, so it is clear that constructive interference in the latter is associated with destructive interference in the polarized rainbow, and this is consistent with the numerical calculations of Khare and Nussenzveig [59]. Note also that this contrast will not be present in the secondary bow because of the two internal reTections in the drop. Near the Brewster angle the Fresnel coeYcient of reTection for parallel-polarized light varies substantially (in relative terms), so that the wave amplitude cannot be considered constant along the wavefront, even near the rainbow angle. This contrasts with the case for perpendicular polarization, for which the Fresnel coeYcient while varying, is never zero and so (in relative terms again) the wave amplitude may be treated as constant to a Arst approximation. This is the basis of the Airy approximation and while not completely satisfactory, provides a reasonable representation of the common rainbow as we have seen. The expression for the cubic wavefront has been noted earlier. A fuller account of its derivation may be found in the book by Humphreys [18] to which we now refer. By considering small deviations from the angles of incidence and refraction corresponding to the D7escartes ray, and with judicious use of Maclaurin’s theorem, Humphreys deduces that in a coordinate system centered on the point of inTection (see Eq. (2.3) above, where h = 3c) y=
h 3 x ; 3a2
where in terms of the order n of the rainbow and the refractive index N 1=2 (n2 + 2n)2 (n + 1)2 − N 2 h= ; (n + 1)2 (N 2 − 1) N2 − 1
(6.5)
(6.6)
a being as before the radius of the drop. For the primary rainbow (n = 1) and N = 1:33, h ≈ 4:9; for the secondary rainbow (n = 2), h ≈ 28. Obviously h is not independent of N . Let IB ; RB and IR be the Brewster angles of incidence, reTection and the angle of incidence for minimum deviation respectively, and furthermore let 5 = i − IR and d = − R , where i and refer to general angles of incidence and scattering, respectively. Then after some tedious but elementary manipulations [18] we may write x = a5 cos IR
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and the integral superposition of the wave amplitude in the direction d as
∞ A= k(x) sin(!t − ) d x = A sin !t − B cos !t ; −∞
(6.7)
where k(x) is the wave amplitude per unit length along the wavefront, ! is the angular frequency of the (monochromatic) light and is the phase shift for a wave in direction d from position x with respect to a wave emanating from x = 0 in the same direction. The expression for is ([18], see also Section 2) 2 hx3 = cos d − x sin d ; (6.8) 3a2 where is the wavelength of the light. We deAne the intensity I = A2 + B2 since (6.7) can be written in the form A = (A2 + B2 )1=2 sin(!t − ,). For the common rainbow, the Airy assumption is that k(x) is uniform throughout (and chosen here as unity without loss of generality), and B = 0 since is an odd function of x, so for small values of d the standard Airy intensity is obtained as A2 where
∞ 2 hx3 cos − xd d x : (6.9) A= 3a2 −∞ The situation is more complicated for the polarized rainbow. For the primary rainbow A⊥ and A|| refer to the amplitudes of light with the E vector perpendicular and parallel to the plane of incidence, respectively. Then by Fresnel’s equations (for unit amplitude incident light) sin2 (r − i) sin(r − i) (6.10) A⊥ = 1 − 2 sin (r + i) sin(r + i) and
tan2 (r − i) A|| = − 1 − tan2 (r + i)
2
tan(r − i) tan(r + i)
:
(6.11)
Note again that A|| changes sign if i passes through IB . If 6 = i − IB ; H = r − RB then using Snell’s law it follows for small 6 and H that H=
6 cos IB 6= 2 N cos RB N
from which k(x) =
A|| 6+H 3 A⊥ cos (IB − RB )
k(x)
1:77 (1 + N −2 )6 ≈ 1:776 = (x + x0 ) ; 3 cos (IB − RB ) a cos IR
or (6.12)
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where x0 = a(IR − IB ) cos IR . Upon substituting all these into the expressions for A and B we And the expected results for the common rainbow, i.e. B = 0 and 2 1=3 a A = 2 Ai(z) ; (6.13) 2 h where 2 xd 2 hx3 ; ; u3 = a2 while for the polarized rainbow 2 1=3 a A = 2 (1:77)(IR − IB ) Ai(z) 2 h uz = −
and 1:77 B = − 2 a cos IR
a2 2 h
2=3
Ai (z)
(6.14)
(6.15)
(the prime referring to derivative with respect to z). The variable z may also be written as 2 2 1=3 4 a d : (6.16) z=− h 2 If the intensity distribution for the common rainbow is expressed in arbitrary units as I1 = [Ai(z)]2 , then for the values for N ( 43 ) and (0:6 m) chosen in [126], the intensity of the polarized rainbow is 2 2=3 2 a 1:77 2 I2 = [1:77(IR − IB ) Ai(z)] + Ai (z) 2 h a cos IR or I2 = 0:0376I1 + 0:232a−2=3 [Ai (z)]2 ;
(6.17)
where z = − 4:92a2=3 d if d is expressed in degrees. The intensities I1 and I2 are shown in Fig. 22 for = 2 a= = 1500 (corresponding to a drop size of ∼0:14 mm for the chosen value of ). The results agree well with the numerical calculations of Khare and Nussenzveig [59]. The shift between locations of maxima for the two polarizations remains present for much smaller drop sizes also [126], although the Airy theory can give qualitative information only in this regime. KRonnen and de Boer also use the asymptotic forms for the Airy function and its derivative to determine an upper bound for the droplet size where the maxima and minima are interchanged for the two polarization directions. For the parameters they choose, the two intensity distributions are in phase if a ¿ 0:6 mm and out of phase if a ¡ 0:6 mm. It is interesting to note from Eq. (6.14) that when IR = IB , i.e. the D7escartes ray is reTected at the Brewster angle, then I2 is proportional to [Ai (z)]2 , and√ is out of phase with the common rainbow at all droplet sizes. This can only occur for N = 2 as noted earlier, but this e9ect is still noticeable for suYciently small drops (i.e. .0:6 mm).
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Fig. 22. The light intensity distribution as a function of − R for both rainbow polarizations: (a) the “common” rainbow and (b) the polarized rainbow. This corresponds, respectively, to the E vector (a) tangential and (b) radial with respect to the rainbow arc. The scattering angle is and the rainbow angle is R . The calculations are for size parameter = 1500 which corresponds to a droplet radius of approximately 0:14 mm in visible light. Note the di9erences in the positions of the supernumerary bows in the two Agures. (Redrawn from [126].)
Returning to the choice k(x) = 1, made earlier for the classical Airy approximation to the common rainbow, it is of course possible to extend that theory by including the variation in the reTection coeYcient for A⊥ also. This in fact contributes only a small correction to the Airy theory: the term in Ai (z) is never dominant. However, the minima in this case are never zero, which is also in agreement with the results reported in [59]. Similar calculations were performed for the secondary rainbow; deviations from the Airy theory are less pronounced, even for the polarized rainbow, where the expression for I2 contains terms in Ai2 (z) and Ai (z) only; since these terms are proportional (see e.g. [129]) the intensity oscillations remain in phase with those of the common secondary rainbow. Finally, the degree of polarization can be easily determined using this formalism. At the rainbow angle d = 0; so from Eq. (6.17) it follows that I2 = R(a) = 0:036 + 0:00465a−2=3 ; (6.18) I1 which for large a compares well with the geometrical optics result [111] of R = 0:039, especially as a → ∞. However, since R as written here increases monotonically as a → 0, for small values of
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a it is more appropriate to consider R as the ratio of the maxima of the intensity distributions for z = 0. By taking values of z equal to −3:25 (common rainbow) and −4:08 (polarized rainbow) as representative maxima (the Arst supernumerary bows) KRonnen and de Boer obtain 2 −2=3 Ai (−4:08) R(a) = 0:0087a = 0:032a−2=3 ; (6.19) Ai(−3:25) R(0:14) = 0:12 (a is in mm) which is to be compared with the prediction of 0:054 from the above asymptotic-type geometrical optics result. This implies that the polarized rainbow is more readily observable for small droplets (e.g. as in fogbows) than is predicted by geometrical optics. 6.3. Comparison of theories There have been several comparisons between some or all of the following: geometrical optics, Mie theory and the Airy approximation. Ungut et al. [43] studied the scattered light from transparent spherical particles with a range of diameters from 1 to 100 m, and in the forward ◦ ◦ scattering angle range 0 –20 ; and compared the results of Mie theory and geometrical optics. For the parameter ranges considered they concluded that the latter is a reasonable approximation to the former, especially if the particles are assumed to deviate from perfect sphericity in any cross-sectional area by a change in the diameter d . More recently, Wang and van de Hulst [42] have given a detailed computational account of an eYcient and accurate method of evaluating the Mie coeYcients for . 50 000, corresponding to drop diameters up to ∼6 mm for visible light. They also compare their Mie theory calculations with the Airy approximation (generalized to an arbitrary number of internal reTections; this number is denoted by p − 1; p ¿ 1, consistent with the notation in [6] and Section 5 above). SpeciAcally they And that for drop diameters ∼ 1 mm there is agreement in all details, including the polarization and the position and intensity of the supernumerary maxima; this agreement is still fairly good down to a diameter of ∼ 0:1 mm. It transpires that the higher the value of the refractive index, the closer is the agreement between the two theories for the primary rainbow (p = 2) for drop diameters as small as 0:02 mm. They also investigate Alexander’s dark band and conclude that in addition to the presence of externally reTected light (p = 1) there is a signiAcant contribution from the p = 6 and 7 rainbows. The positions of these are very sensitive to the wavelength of the light. We note some other points from this paper, the Arst being of historical interest. The authors point out that since 1957 (when Van de Hulst’s book [7] Arst appeared) the large gap that existed between small -values, for which Mie theory was computationally feasible, and large
-values, for which the Airy theory is valid, is much closer to being closed today. This gap was 30 . . 2000, and since that time (as detailed earlier) the complex angular momentum theory ([10] and many references therein) and the development of highly eYcient computational schemes have helped to close the gap (from the upper and lower ends, respectively). Now, reliable Mie computations are feasible for up to at least 50 000. Details of these can be found in [42]. We have alluded earlier to the curvature term h=3a2 for the cubic wavefront approximation of Airy (see Eqs. (6.5) and (6.6)), where h is expressed in terms of the order of the rainbow and the refractive index. Note that the curvature of the wavefront is greater for higher values of p (and hence n) and smaller values of a (or, for light of a given wavelength, for smaller values
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of ), so that the angular spacing between two successive supernumerary peaks increases with increasing p (or n) and=or with decreasing a. In both [7] and its relatively recent extension [42] an expression is given for the rainbow peak intensity: in particular there is a size dependence of ∼ x7=3 which is consistent with an amplitude dependence of ∼x7=6 in Eq. (2.4) of [61], so there is no discrepancy in this regard between the two approaches, i.e. between the Airy theory and the complex angular momentum theory. For a drop with diameter 3:2 mm, the authors also found excellent agreement between the Mie and the Airy theories (such a large drop would not, of course retain its spherical shape in the atmosphere). The primary rainbow was found to be ≈ 7:7 (red light) to ≈ 8:4 (violet light) times more intense than the secondary rainbow, and all the Airy maxima up to about the tenth supernumerary bow were at the predicted angles and had the correct intensities. Also the angular separation of the rainbows was found to decrease as the light shifted from violet to red: the shift in the secondary (p = 3) being twice that in the primary (p = 2) rainbow. For a drop diameter of 0:1 mm there is still (surprisingly, as the authors point out) good agreement between the Mie and Airy intensities for the Arst maxima in cases p = 2 and 3. Amongst other features, the dark band is still seen in the violet but diminishes in the red, and the Airy theory predicts somewhat larger shifts in the higher supernumerary bow positions for p = 2 and 3. Tests for even smaller sizes (and various refractive indices) indicate that the Airy theory predicts well the features of the Arst peak of the primary rainbow. In the dark band, where the intensity may drop to below 0:5% of that in the primary peak, the intensity is well explained as contributions from the p = 0 (external reTection) and the p = 6 and 7 rainbows. The overall conclusion drawn by Wang and van de Hulst is that the Airy theory provides simple and reliable information over a wide range of sizes, and furthermore this may be useful in determining properties of particles in laboratory, meteorological or astronomical experiments. Related to these results is a set of observations made earlier by Sassen [1] in connection with angular scattering and rainbow formation in pendant drops. These are laboratory-produced near-spherical drops with certain similarities to the shape of distorted raindrops. The latter become increasingly oblate and unstable with increasing size, and tend to orient themselves uniformly in space as they fall. Measurable raindrop distortions have been noted for drops as small as 0:3 mm in diameter [130] and are signiAcant for diameters &1 mm. As might therefore be expected, rainbow features often vary with the position of the generator of the rainbow “cone” as the observer views di9erent parts of the rainbow. This is because of the change in geometric cross-section of the drop as the angle of observation changes (for more on this see Section 6.4). Such variations have been observed [131]. Obviously in any given situation it is expected that there will be a distribution in raindrop size. It is frequently the case that rainfall from convective showers (conditions under which rainbows are frequently observed) contain “supermillimeter” drops, so information on near-spherical drops may well help to explain occasional anomalous features (i.e. unlikely to occur for purely spherical drops) such as reports of tertiary rainbows ◦ [14,132–134] at 42 (while standard theory predicts their existence, they are considered too faint and too close to the sun’s position to be observable, although these reports might refer to reTected or reTected-light rainbows [4]). The pendant drops discussed in [1] are more akin in shape to the familiar “teardrop” shape so beloved of cartoonists. Nevertheless, they combine spherical and non-spherical scattering mechanisms that are considered relevant to those for aspherical raindrops for the following
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two reasons: (i) they retain a circular cross section in a horizontal plane and (ii) display a reduction in surface curvature (compared to a spherical drop) along the vertical sides of the drop. These are exactly the same features displayed by the “hamburger-bun” shapes of larger raindrops (see further references in [1] for details of numerical and theoretical simulations of realistic raindrop shapes). Angular scattering experiments revealed that such near-spherical drops generate exceptionally well-deAned primary, secondary and associated supernumerary bows, but unexpectedly strong higher-order bows as well, perhaps because of the reduced curvature in the central elongated regions, thus providing relatively more surface area within the drop available for generating rainbow rays. As with the later work of Wang and van de Hulst [42], it was found that the precise angular positions of at least the Arst 10 intensity maxima for the primary and secondary rainbows were predicted by the Airy theory. Interestingly, it is claimed that the theory fails to account for the position of the sixth-order rainbow (p = 7), but in [42] it was found that this rainbow, along with that for p = 6 is found just inside Alexander’s dark band. Dave [135] presented results of Mie computations for large non-absorbing spheres and compared them with those corresponding to geometrical and physical optics (i.e. including di9raction e9ects). This was done for monochromatic radiation of wavelength = 0:4 m and water spheres of radii a = 6:25; 12:5; 25 and 50 m, corresponding to size parameters, respectively, of = 31:25 ; 62:5 ; 125 and 250 , well below the region of applicability (and prior to the emergence of the complex angular momentum theory) of the classical ray and Airy theories [7]. Dave considered three regimes using geometrical optics: forward scattering, the rainbow region and the glory region. An approximate value for the scattered intensity in the Arst regime can be found by replacing the sphere by a circular disk of radius a [7]. This unpolarized di9racted radiation has intensity 2 4 J1 ( sin ) If =
;
sin relative to the incident radiation intensity. The maximum intensity of 4 =4 occurs of course at ◦ = 0 ; successive minima and maxima occur as increases away from zero. If the amplitudes of the reTected and refracted are comparable with that of this di9racted component, there can be a signiAcant increase in intensity and interference between these components. (This is sometimes referred to as “anomalous di9raction” mentioned earlier; further details may be found in [7].) Similar interference phenomena may of course occur in the rainbow region on either side of the D7escartes ray of minimum deviation, if two such rays emerge in parallel directions—leading at this level of description to an explanation of the supernumerary bows. Apart from van de Hulst’s suggestion of surface wave involvement as responsible (in part) for the glory, there was at that time no satisfactory detailed account of this phenomenon (the work by Nussenzveig was published shortly after [135] appeared). This paper [135] provides many Agures and related results; it was concluded on the basis of the four size parameters studied that ray optics and Mie theory agree well only if ∼ 800 (and presumably & 800, though Dave’s largest value was = 785:4). He concluded that a satisfactory explanation of the glory should await a more substantial understanding of surface waves and the interaction between radiation Aelds (true), and noted also the lack of a simple explanation for (i) anomalous di9raction in the forward ◦ direction (for ¿ 2 ) and (ii) the appearance of several distinct maxima outside the regions of primary and secondary rainbows.
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The four approaches to the theory of the rainbow discussed so far are, in historical order: geometrical optics, Airy’s approximation, Mie theory and complex angular momentum theory. Crudely, the Arst of these is independent of size of the drop, but only valid for large size ◦ parameters, e.g. & 5000. Airy’s approximation is valid for & 5000 and for | − R | . 0:5 ; the Mie solution is exact but requires the summation of a number of partial waves greater than
(this is no longer the problem it used to be for & 100 as noted above in connection with [42]). The complex angular momentum theory provides a very good substitute for Mie theory if & 50 (in fact it is still very accurate for & 1 [10]) but is mathematically very technical. It was with these restrictions in mind that Mobbs [136] developed another theory, based on Huygens’ principle (as is Airy’s theory) in which he states that the physics is straightforward and only simple computation is required for a substantial range of drop sizes. A virtual wavefront leaving the drop after one internal reTection is examined, as in Airy’s theory, and a set of equations is developed which completely deAnes this wavefront. Mobbs also deAnes, in a similar manner, a virtual wavefront for the externally reTected ray. The distributions of amplitude (for both polarizations) depend on the drop geometry, the wavefront and the Fresnel amplitude coeYcients. Sixteen Fresnel zones were constructed across the wavefront, each of which has an amplitude (uniform across the zone) characteristic of the total light Tux passing through that zone. Thus the wavefronts consisted of 16 annular rings around the (obvious) central axis (z) of the drop; the boundaries corresponding approximately to equally spaced values of the angle of incidence. The basic mathematical features of this approached will be brieTy summarized here (and in the opinion of this author, with regard to Mobbs’ earlier comment about the CAM theory, his calculations are not entirely without mathematical complexity either!). Suppose that the nth zone, 1 6 n 6 16, lies in the angle of incidence interval [in ; in+1 ]. The area orthogonally presented to the incident light between these angles is readily shown to be an =
a2 (cos 2in − cos 2in+1 ) 2
and the area of the zone is, in a cylindrical polar coordinate system, 2 1=2
in+1 dz dR An = 2 R 1+ di : dR di in
(6.20)
(6.21)
DeAne
Gp; n =
an
Ap; n
1=2
;
where the subscript p refers to externally reTected light (p = 0) and internally reTected light for the primary rainbow (p = 2). The rainbow rays undergo two refractions (transmissions) and a single reTection. So using the Fresnel amplitude coeYcients the amplitude corresponding to the nth zone is given by (1)
A2; n = G2; n
sin(2i) sin(2r) sin(i − r) sin3 (r + i)
(6.22)
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for polarization 1 (magnetic) and (2)
A2; n = G2; n
4 sin(2i) sin(2r) tan(i − r) [sin(2i) + sin(2r)]2 tan(r + i)
(6.23)
for polarization 2 (electric). The corresponding amplitudes for externally reTected light are (1)
A0; n = − G0; n
sin(i − r) sin(i + r)
(6.24)
and (2)
A0; n = G0; n
tan(i − r) : tan(i + r)
(6.25)
Mobbs then applies Fresnel’s formulation of Huygen’s principle by examining the path di9erence between rays that meet the wavefronts such that 1 = − R . The disturbance from an element of area d S of the wavefront near a point P, a large distance s from the drop can then be expressed in terms of the A coeYcients above, some complex exponential factors in time and space, and an obliquity factor, details of which may be found in [136]. By suitable formal integrations, using results of the type [105]
cos(n − x sin ) d = Jn (x); n = 0; 1; 2; : : : (6.26) 0
and after considerable reduction, the amplitude coeYcients Sp( j) (; ) can be written in terms of the rather formidable expression Sp( j) (; ) =
16 n=1
−
Rp; i
A(p;j)n ( −1 tan 1)e−iklp sin 1 [Rp J1 (kRp sin 1)eikzp cos 1 ]Rp; in+1 n
16
A(p;j)n (k −1 )e−klp sin 1
in+1
in
n=1
d Rp ×eikzp cos 1 J0 (kRp sin 1) di di
Rp
1+
d zp d Rp
2 1=2
+ sec
;
(6.27)
where the path di9erence calculated is between light from the point (lp ; 0; 0) and the point P(Rp ; ,; zp ). The addition of the contributions from the internally and externally reTected light is carefully considered in terms of the phase di9erence between the two; then the total amplitude coeYcients S (j) (; ) are S (j) (; ) = Re(S2( j) ) + Re(S0( j) ) cos
( j)
− Im(S0 ) sin
+i[Im(S2( j) ) + Re(S0( j) ) sin
+ Im(S0( j) ) cos ] :
(6.28)
Concerning the validity of this approach, Mobbs makes an interesting point: its validity is based on the assumption that the wavefronts can be determined by constructing rays (orthogonal
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Fig. 23. The light intensity distribution as a function of for polarizations 1 and 2 (excluding externally reTected light) for = 1500, in which the results of the complex angular momentum theory and Mobbs’ theory are compared. Agreement for polarization 1 is excellent and the di9erences for polarization 2 are small. (Redrawn from [136].)
to the wavefronts), and on this basis Tricker [8] has attempted to determine the validity of the Airy approximation. In fact his argument uses the Fresnel zone concept also, but Mobbs points out that the fundamental assumption in the Airy theory is (instead) that the amplitude is uniform across the wavefront. However, his result is applicable here: the radius of the drop should be large compared with 4 , or 8 . In practice, Mobbs also found an upper limit to determined by the diYculty of evaluating the sixteen integrals for large values of . For ◦ ◦ the range of scattering angles examined, namely 136 6 6 142 , the agreement between this theory and the complex angular momentum approach is very good for = 500 and 1500 (the latter excluding externally reTecting light, which superimposed small rapid oscillations on the main features of the internally reTected light; see Fig. 23). Even for = 50 the agreement is fairly good for both polarizations. This approach represents an interesting alternative to the complex angular momentum theory, and a feasible one when surface waves and higher-order reTection terms are not important; however the power, generality and beauty of the latter theory (not to mentions its broad range of accuracy) cannot be refuted, even if it is considered, in some applications, to be the proverbial “sledge-hammer cracking a nut”. Lynch and Schwartz [137] (see also [138]) used Mie theory to study the optics of rainbows and fogbows (i.e. rainbows formed in clouds and fogs, as opposed to rain showers) for monodisperse drop radii in the range 3–300 m (for light of wavelength ∼ 0:5 m this corresponds to 38 ¡ ¡ 3800). This upper limit was chosen because of the tendency of larger drops to oscillate or become aerodynamically distorted. Furthermore, drop sizes below 3 m exhibit no ◦ ◦ discernible rainbow. The range of scattering angles in the calculations was 110 ¡ ¡ 160 . In their calculations they included a realistic solar illumination spectrum (as modiAed by the earth’s atmosphere) and the Anite angular size of the sun. Fogbows tend to be paler than rainbows (being formed from smaller drops, a ∼ 50 m), and are also wider; the maximum intensity ◦ occurs a few degrees higher than the 138 scattering angle for the primary rainbow. Amongst other goals, the authors were interested in obtaining a quantitative understanding of the transition between rainbows and fogbows. They calculated the scattering intensity function I () from an appropriate integral involving the ratio of the scattered-to-incident radiation for unpolarized
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light; this latter quantity is the scattering matrix element S11 (a; ; ) in their notation. (To use their results to model a rainbow formed in a rain shower would require I () to be weighted with the drop distribution n(a) and an additional cross-section factor.) In particular, for the ranges considered the authors found that: (i) the rainbow attains a minimum angular width for a = 100 m. Indeed, as a decreases the widths of both primary and secondary bows increase, their contrasts (as deAned in [137]) decrease and their scattering angles of maximum brightness move away from Alexander’s dark band. (ii) the supernumerary bows for the primary rainbow attain maximum visibility for a = 50 m; they grow wider and further apart as a decreases. For values of a above about 56 m their ◦ separation is less than the sun’s angular diameter of 0:5 and their amplitudes are greatly reduced as a result of the concomitant smoothing e9ect; below about 36 m the wider bows lose contrast. There are other factors inTuencing this of course, such as drop size, shape and distribution [139,140]. ◦ (iii) Alexander’s dark band maintains a constant angular width of about 9:5 , apparently because the primary and secondary bows, while widening grow away from the dark band with decreasing drop size; its angular width, however, is a function of drop size; (iv) the maximum linear polarization of both types of bow is also a function of drop size, reaching a maximum of 90% for the primary bow and 50% for the secondary bow when 20 m ¡ a ¡ 100 m, and (v) the secondary fogbow is observable only when a ¿ 10 m. The authors further suggest that a fogbow be deAned as any rainbow formed in drops for which a ¡ 35 m. In 1983, Fraser published a paper with the intriguing title “Why can the supernumerary bows be seen in a rainshower?” [141]. In order to see why there should be such a question in the Arst place a little background will be useful. The supernumerary bows are essentially an interference phenomenon, as Young pointed out in the Bakerian Lecture of 1802 [142]. The Airy approximation predicts, as we have seen, that the spacing between adjacent bows, and the width of the primary bow should increase with decreasing drop radius. Since in reality however, normal rainshowers have a spectrum of drop radii (spanning more than an order of magnitude [143]) it would seem that no consistent pattern of supernumeraries should be seen, contrary to what is frequently observed. These bows form near the top of the bow, and so any explanation of their existence should incorporate the fact that the light in this portion passes through the vertical cross section of often-distorted drops. While the standard ray approach (supplemented by the wave concept of interference) is a perfectly reasonable way to explain the existence of supernumerary bows for two sets of rays emerging in the same direction from monodisperse drops, Fraser points out that “a more compelling visualization is obtained if the rays are replaced by waves”, and this is certainly the case (see Fig. 4). The ray picture cannot of course account for di9erent drop sizes, whereas the wave picture does, and indicates that large drops yield tightly spaced supernumeraries with narrow maxima, while small drops yield broader maxima with wider spacing. Fraser uses the Airy approximation in his calculations, pointing out that it does adequately identify the positions of the maxima for drop radii in excess of 0:05 mm (50 m), which is certainly outside the fogbow range according to [137]. The angle of minimum deviation is a9ected by drop distortion: MRobius showed that the deviation angle of the D7escartes ray would in fact be increased [144].
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Fraser used ellipsoidal drops in his calculations, noting that for the smaller drop sizes of interest to him, this is an excellent approximation. It appears that the minimum angle of deviation is also weakly dependent on solar elevation (presumably because of the di9erent-sized drops that ◦ contribute to the light distribution at the top of the rainbow), but only one such elevation (0 ) was used since this e9ect is negligible here. Since smaller drops yield larger angles for the D7escartes ray, it is apparent that there is an intermediate drop size that will yield a minimum deviation angle D, i.e. in formal terms D(a) = min{min D(i)} a
i
and it these drops which cause the supernumeraries that are observed. This drop size turns out to be in the range 0.1–0:2 mm for the primary bow and in the range 0.2–0:3 mm for the Arst two supernumeraries. No other drops make a signiAcant contribution. On the basis of his results Fraser predicts that if supernumerary bows are seen in a rain shower, their separation ◦ will be ∼ 0:75 , independent of the particular size distribution of the raindrops, and found from ◦ photographs typical separations of ∼ 0:70 : In addition, Fraser [145] has pointed out that the explanation by Voltz [150] for the absence of supernumerary bows near the base of the primary rainbow (when the sun is low in the sky) also applies to the intensiAcation of the red light contribution in the same location. (He also states that the rainbow is a Ane example of a phenomenon in which theory tends to color observations.) Near the top of a rainbow the red color is much less intense that towards the base, i.e. the more nearly vertical portions of the bow. It is only in these regions that the incident light, which is deviated in a more nearly horizontal plane, encounters a circular cross section in a Tattened drop. Large drops will still contribute to the rainbow, whereas such drops in the vertical plane displace the rainbow angle inward toward the antisolar point because of their distorted cross section in that plane (this was in fact Arst demonstrated by MRobius [144]: elliptical drops lower the top of the bow. Voltz estimates that drops smaller in radius than 0.25 – 0:5 mm are suYciently spherical to contribute to all portions of the rainbow; larger than this they will contribute only to the vertical portions, which in turn explains why the rainbow is brightest there—the larger drops substantially increase the intensity—and why the supernumeraries are absent—the intensity maxima overlap. The complete spectrum of “colors of the rainbow” may only occur near the foot of the bow (along with the pot of gold, etc.). 6.4. Non-spherical (non-pendant) drops Subsequently, and perhaps inspired by the work of Fraser, KRonnen discussed the same type of phenomena in more mathematical detail, but speciAcally for the secondary rainbow [147]. They (i.e. supernumeraries for the secondary bow) are seen only on extremely rare occasions [148]. KRonnen points out that the D7escartes ray for the secondary rainbow shows very little dependence on the drop oblateness (for visible light) due to the fact that oblateness-induced changes in the two refractions and two interior reTections essentially cancel out, and so in view of Fraser’s explanation for supernumeraries associated with the primary bow, they are unlikely to be seen for the secondary bow (for most solar elevations there is now no stationary point for D(a)). ◦ However, for solar elevations exceeding about 35 the Tattening of the drops can result in a
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minimum for D(a), and this may be suYcient to cause the Arst supernumerary—particularly for red light—to become visible (see [147] for a detailed explanation). KRonnen approximates a falling drop by an oblate spheroid, with its two semi-major axes of length a horizontal and its semi-minor axis of length b vertical. The equivalent (i.e. same volume) spherical drop radius is r = (a2 b)1=3 and for r ¡ 1:2 mm this is a reasonable approximation (and an excellent one if r ¡ 0:5 mm). Following MRobius [144] the drop ellipticity is deAned as G=
a−b : a+b
(6.29)
For drops falling at terminal velocity (assumed here) Green [149] derived an approximate relationship between the ratio b=a and r. It is given by ba−1 = [(4=17)(17B=4 + 1)1=2 + 13=17]−3=2 ;
(6.30)
where B = G r 2 g =/ is called the Bond number; G ; g and / are respectively the drop density, e9ective gravity (g = (G − Gair )G−1 g) and surface tension coeYcient. By substituting this into the expression above for G and expanding in a Maclaurin series about r 2 = 0, KRonnen obtains the further approximation G 0:05r 2 if r is expressed in millimeters. He states that this simple result is useful for r ¡ 1:5 mm. According to [144] the deviation between the rainbow angles for spheroidal and spherical drops is ]R ≡ R (spheroid ) − R (sphere) ; i.e. ◦
]R =
180 ◦ 16G sin cos3 cos(2h − 42 ) ;
(6.31)
where here is the angle of refraction of the D7escartes ray for spheres, h is the (angular) solar ◦ ◦ elevation and 42 (= 180 − R (sphere)) is the angular distance from the bow to the antisolar point. (KRonnen also notes a sign error, and signiAcant consequences therefrom, present in [152]). For angles of observation other than for the top of the bow, (i.e. a non-vertical scattering plane) ◦ the corrections to G and h are also provided in [147]. In the visible range the angle ( 40 ) ◦ varies by only about 1 , so equation (6.31) is almost independent of wavelength. Furthermore, it is consistent with Fraser’s conclusions that ]R is a strong function of drop radius r but ◦ only a weak function of h (h ¡ 42 ). Thus to a good approximation, ◦
◦
]R = 13 r 2 cos(2h − 42 ) ≡ Cr 2 ; ◦
(6.32)
◦
where 10 6 C 6 13 : The result corresponding to (6.31) for the secondary rainbow is ◦
180 ◦ ]R = − 64G sin cos3 cos 2 cos(2h − 51 ) ;
(6.33) ◦
◦
where the negative sign results from changing the scattering angle interval to [180 ; 360 ], which enables the results for the primary and secondary bows to be presented in a more convenient ◦ and consistent manner. Note that in this case ]R is zero when = 45 ; it is a straightforward
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matter to show from geometrical optics that cos 2 =
5N 2 − 9 4N 2
and so this occurs when N = 9=5 1:342, which for water corresponds to violet light. This of course means that now there is a strong dependence of ]R on wavelength for light this wavelength, which is readily formulated using di9erentials. It turns out to be necessary to include terms up to G2 for the secondary rainbow, the result corresponding to (6.32) above being
]R = C1 r 2 + C2 r 4 ;
(6.34)
where the coeYcients C1 and C2 are deAned in [147]. The author further demonstrates that the stationary behavior for the secondary rainbow near its Arst supernumerary occurs if r 0:7 mm; this in turn leads to a yet simpler form for (6.34), namely ]R = C r 2 , where C = C1 + 0:5C2 : C is a strong function of wavelength (via C1 ) and solar elevation (via C2 ). For falling drops not exceeding about 1:8 mm in radius (G ≈ 0:1), the Airy theory can still be applied if the rainbow angle for spheres is replaced by that for spheroids using the expression for ]R : For larger eccentricities this is not the case, because the Airy intensity distribution (which may be thought of as corresponding to a fold di;raction catastrophe) may be transformed into a hyperbolic umbilic di;raction catastrophe in any given drop of large enough G. This is because two additional rays with skew paths through the drop can also contribute to the rainbow interference pattern [151]. (Di9raction catastrophes will be discussed in more detail in Section 7). This has been observed to occur for the primary rainbow when the scattering plane is approximately horizontal [147]. For reasons discussed in that paper, it is conjectured that no such transformation occurs for the secondary rainbow (though it may occur for other refractive indices). Furthermore this e9ect can be neglected if there is a broad distribution of drop sizes, because for the Marshall-Palmer distribution [152] (for example) less than 1% of the drops exceed 1:5 mm in radius, so the e9ect is swamped by that from the smaller drops. Returning to the regime for which the analytic form from the Airy theory is still appropriate, the following expression arises for the intensity distribution near the secondary rainbow angle ◦ R ( 231 ) I (r; ) ˙ r 7=3 Ai2 [f(r; ; )] ; where f(r; ; ) (not to be confused with the scattering amplitude) is deAned as: 500r 2=3 f(r; ; ) = − 3:1 [ − R (sphere) − Cr 2 ] :
(6.35)
(6.36)
The same functional form applies to the primary rainbow also with some di9erences in the argument of the Airy function. The above argument is stationary at r = rs where 4Crs2 = − R (sphere) : If C ¿ 0, the stationary behavior of f is in the oscillatory region of the Airy function, which means that the integral of Eq. (6.35) over a broad drop-size distribution also has an oscillatory
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component, and the spacing between consecutive maxima in the intensity distribution can be inferred from this. The angular spacing 1; 2 ( ; h) between the Arst and second supernumerary bows is given in [147] by the expression 1=2 ◦ 1=4 1; 2 ( ; h) = 0:71 C ( ; h) : (6.37) 640 ◦
◦
For the primary bow with C = 13 ; 1; 2 = 0:88 at = 640 nm, consistent with the results of ◦ ◦ Fraser [141]; for the secondary bow with C = 1 at the same wavelength, 1; 2 = 0:71 : In order to simulate the intensity distribution of a typical secondary bow produced by a rain cloud, KRonnen assumed the drop-size distribution to be of the Marshall-Palmer type [152] dn ˙ exp(−6r) ; dr where n is the number of particles of radius r (in mm). This relation was then multiplied by (6.35) and integrated over both r and the solar disk. Amongst KRonnen’s conclusions were the following: (i) the intensity of the main peak of the secondary bow is a factor of 4 greater at the top than at the base; (ii) at low solar elevations, no supernumeraries (for the secondary bow) should ever be observed; and (iii) at higher solar elevations the Arst supernumerary should be observable, especially at the red end of the visible spectrum if a Alter is used. The primary reason for the diYculty of observing the supernumerary bows is that the secondary rainbow angle is much less sensitive to the Tattening of rain drops than is the corresponding angle for the primary bow. At low solar elevations, an increase in drop oblateness shifts the secondary rainbow pattern toward smaller deviation angles, which is in the wrong direction for producing supernumeraries. At higher solar elevations this is not necessarily the case, but even then the background light may swamp an otherwise visible supernumerary bow. Lock [153] discusses the observability of supernumerary bows and atmospheric glories, pointing out that one of the chief factors inTuencing this is the spatial coherence width of sunlight as received at the earth. Sunlight in fact possesses little spatial (or temporal) coherence, and this places limits on both the number of supernumerary bows that can be produced adjacent to the primary (for a given size of drop) and on the size of drops that can produce a glory. It is shown by Wood [154] that the diameter of the area over which the phase is constant is d ≈ R =2r for an object of apparent diameter 2r=R (r being the actual radius if circular) observed at a distance R. For sunlight with ≈ 0:5 m, d ≈ 50 m, so if the distance between the incident light rays exceeds d then the lack of coherence results in no interference phenomena, i.e. supernumeraries or glories. The idea behind Lock’s investigation is to test the consistency of this approach with the observations by calculating the optical path di9erence between rays exiting the drop parallel to each other. According to these calculations, only the Arst two or three supernumerary bows have incident rays that are separated by less than the value of d given above (in fact Lock uses d ≈ 40 m, so only the Arst supernumerary bow meets his criterion). Interestingly, the intensity distribution calculated for scattering by a sphere of radius 250 m with partial coherence qualitatively resembles the pattern for di9raction by an edge, as opposed to the standard Airy pattern for coherent radiation (cf. Figs. 1 and 2 in [153]). As far as the glory is concerned, Lock demonstrates that the peak observability of the glory occurs for droplets with radii in the range 10 –20 m. Some pertinent comments are made
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about the nature of the surface wave e9ect in this phenomenon that are usefully stated here. It is known that the wave-interference factor for glory backscattering (exiting the drop as a toroidal wave front) is [7,50] 2 2 r 2 2 r F = (c1 + c2 ) J0 + (c1 − c2 ) J2 ; (6.38) z z at a distance z behind the scatterer, r being the radial coordinate. The quantities c1 and c2 are the products of the transmission and internal reTection coeYcients for the incident transverse magnetic and electric Aelds, respectively. (There is also a geometric scaling factor 2 a ]r ˙ a4=3 associated with the edge region ]r of the drop.) As noted earlier, for water drops there must ◦ be a surface wave traversing a 14 arc to account for the glory, so the coeYcients c1 and c2 must also include terms to account for the creation and propagation of surface waves, and so they will not be constants in fact, for they will depend on the droplet size. It is shown in [47] that damping of surface waves increases with drop size, with c1 ≈ c2 ˙ exp(−0:4 1=3 ) : For a perfectly coherent light source the contribution to the intensity of the glory from the surface wave is proportional to [47] 8=3 exp(−0:8 1=3 ) which has a relative maximum at = 1000 or a ≈ 88 m. This may be understood as follows: If a increases (below this maximum value), the geometric scaling factor likewise increase and this allows more rays to contribute to the glory scattering. Above this maximum value of a the damping of the surface waves reduces the contribution [153]. Based on estimates of Van de Hulst [7], Lock uses in his calculations the forms c1 = − 0:2 exp(−0:4 1=3 ) and c2 = 1:0 exp(−0:4 1=3 ): Again, signiAcant quantitative di9erences are found for the case of partial coherence versus coherence in the spatial domain, and it appears that in the presence of a wide distribution of drop sizes the glory is primarily produced by a small range (10 –20 m). Since many rainbows are formed in association with thunderstorms, it is natural to enquire as to the e9ects on them of strong (vertical) electric Aelds. A study by Gedzelman [155] does just that, noting that any force that can alter the shape of a raindrop will also a9ect the rainbow. Indeed, rainbows have been observed to quiver during thunderstorms [148]. A vertical electric Aeld is known to polarize the drops and stretch them vertically [156,157] which of course is the opposite e9ect to that of aerodynamic forces on falling drops. The extremely large vertical Aelds in the vicinity of thunderstorms (E & 105 V m−1 ) may produce signiAcant and hence measurable changes on drop shape. Any net charge present on the drops also a9ects their shape by reducing the e9ective surface tension, thus enhancing any propensity to depart from sphericity, though this is realistic only for the largest drops, and since these do not contribute to the supernumeraries at the top of the bow, electrical charge should not a9ect the visible appearance of the rainbow signiAcantly. If electric charge is neglected, the formula used in [155] for the eccentricity of a raindrop in the presence of a vertical electric Aeld of intensity E is (on correcting a minor typographical error) e=
a − b 3r(2Ggr − 3jE 2 ) = ; a+b 16/
(6.39)
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331
where r is the drop radius, G is the e9ective density for water in air, H is the permittivity and / is the surface tension of water under these conditions. The quantities a and b are, respectively, horizontal and vertical semi-axes. This formula is valid only for small eccentricities and hence small drops (see [156] for a more general formula for e) because it neglects the progressive Tattening of the base for larger drops; nevertheless it is a reasonable approximation for drops of radius . 1 mm. Note that drops with radii smaller than 3HE 2 =2Gg will be prolate spheroids (e ¡ 0); for E = 4 × 105 V m−1 this implies r = 0:2 mm. Fields of this magnitude are only to be found in thunderstorms in general; the typical fair-weather value is about three orders of magnitude smaller and will be unlikely to alter the visible rainbow. Gedzelman performed a similar set of calculations to KRonnen [147] for non-spherical drops, including solar elevation and drop size distribution, and integrating over the angular width of the solar disk. He concluded that the electric Aeld elevates the top of the bow by decreasing the scattering angles of the Arst three Airy maxima (because of reduced drop oblateness); it also Tattens the minima in these scattering angles, thus producing a brighter bow because it increases the e9ective range of drop sizes contributing to the bow. However, it also decreases the intensity contrasts of and spacing between the supernumeraries near the top of the rainbow. Shipley and Weinman [158] carried out a detailed numerical study of scattering by large dielectric spheres, noting that their average light scattering properties can be adequately described by di9raction theory in the forward direction and geometrical optics at most other angles. They compared their results for N = 1:333 and size parameters 200 6 6 4520 using Mie theory with the extinction eYciency approximation of Van de Hulst [7], the standard forward di9raction formula and with geometrical optics. Excellent agreement was found for the Arst two comparisons, and as expected for the case of geometrical optics, agreement was found with the exact averaged Mie calculations (for ∼ 4520), but neither the forward di9raction peak nor the backward glory can be predicted (they did not discuss the rainbow problem in their paper). They investigated also the magnitude and angular structure of the glory, verifying both the results of Bryant and Cox [54] and the predictions of Nussenzveig concerning the backscatter oscillations and phase function oscillations (three predicted oscillations were veriAed, and a fourth was discovered for 200 6 6 2000). Liou and Hansen [159] carried out a similar set of comparison calculations, but this time for both non-absorbing and absorbing spheres for several refractive indices in the range 1:16Re N 6 2:0. They were particularly interested in the intensity and polarization for single (as opposed to multiple) scattering by polydisperse spheres for use in cloud microstructure studies. For Re N = 1:33 and 1:50 they determined that the ray theory and Mie theory are in close agreement if the size parameter & 400. 6.5. Rainbows and glories in atomic, nuclear and particle physics “Do we have enough imagination to see in the spectral curves the same beauty we see when we look directly at the rainbow?” [207] While the majority of this review is devoted to the mathematical physics of the optical rainbow and glory (albeit with signiAcant quantum mechanical connections, such as tunneling),
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a synopsis of rainbows and glories ‘writ small’ is entirely appropriate. A useful summary can be found in Chapter 16 of [10], where the following three arenas are noted: (i) In atomic and molecular scattering, observed e9ects include di9ractive and rainbow scattering, glories, orbiting and shape resonances; (ii) In nuclear heavy-ion scattering, semiclassical e9ects such as nuclear rainbows, backward and forward nuclear glories, and surface waves in di9ractive scattering. (iii) In particle physics, one type of phenomenological model based on the tunneling of surface waves is very consistent with observations of high-energy proton–proton scattering. A summary of the literature in these three contexts up to about 1991 can be found in [10]; more recent “key” references (where many further relevant citations are to be found) will be identiAed below. A common potential used to describe the interaction between two neutral atoms is the Lennard–Jones (12; 6) potential r 6 rmin 12 min V (r) = V0 −2 ; r r where V0 is the depth of the well and rmin is the position of the potential minimum; this describes a repulsive central core within a long-range attractive potential. In addition to derivations of CAM representations of the scattering amplitude [160 –163], Regge poles and trajectories have been computed for several interatomic potentials [164,165] (see also related earlier work, previously discussed [19,68,166 –168]). The Arst atomic rainbow was observed in 1962 by Beck [169]. In [170] di9erential scattering cross sections were studied; the main peaks correspond to the (primary) rainbow and several supernumeraries. By contrast to the optical rainbow, where the (geometrical-optic) rainbow angle depends only on the refractive index N; there is signiAcant ‘energy dispersion’ in the atomic case because of the energy (k 2 ) dependence of V (r) Natomic = 1 − 2 : k The results are like spectral curves for di9erent colors; hence the reason for the comment at the beginning of this subsection. Note also that the size parameters are also functions of energy. Thus scattering can be used as a probe of the depth V0 of the attractive well in the interatomic potential. Considered as di9raction features, the supernumeraries (which depend on the size parameter) provide information on the range of the potential. But things get even better: there are also ‘rapid quantum oscillations’ arising from interference between attractive and repulsive trajectories. These probe di9erent parts of the potential so that, cumulatively, they can provide information on the shape of the potential [171–173]. Furthermore, rainbow e9ects have been detected in rotational and vibrational excitations in various contexts [174,175]. Forward glory paths and interference oscillations also exist, as do glory undulations (observed in the total cross section for several noble gas systems [176]). Their spacing is a measure of the strength-range product V0 rmin [177]. There is also a physical interpretation for the total number of glory oscillations [178,179]: since the phase di9erence between successive maxima is ; and this approaches the s-wave phase shift as E → 0; the total number of oscillations is the same (semiclassically) as the number of multiples of contained in the zero-energy phase shift. It follows from Levinson’s theorem [180,92], that this is equal to the number of bound states of the potential (see [181] for improved estimates of glory undulations based on a uniform
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approximation). The addition of a centrifugal barrier can induce shape resonances, as noted for the optical rainbow [182]; orbiting resonances also play a role in atomic scattering [183– 185]. A major aspect of atomic and molecular scattering experiments is of course to use the observational data and relevant mathematical techniques to solve the ‘inverse problem’, i.e. to reconstruct the original potential [71,172,173]. Similar types of remarks apply to nuclear heavy-ion collisions; semi-classical techniques are developed in this context by Brink [186]. However, in contrast to atomic collisions, the Coulomb interaction must be incorporated into the theory (and the background integral is no longer small [187]); allowance must also be made for strong absorption for distances less than a critical radius Rs (called the strong absorption radius) Nuclear interactions are commonly described via the complex Woods–Saxon optical potential V (r) =
V0 iW0 − : 1 + exp[(r − Rr )=ar 1 + exp[(r − Ri )=ai
The presence of the imaginary part describes absorption from the elastic channel; Re V (r) describes a smooth potential well of depth O(V0 ) and depth O(Rr ); ar is a parameter describing ‘surface di9useness’. Similar interpretations apply to Im V (r): When Ri ¡ Rr the potential is said to be surface transparent. The e9ective potential for a given angular momentum is composed of the long-range repulsive Coulomb potential (dominant at large distances), the centrifugal barrier and the complex nuclear optical potential V (r): (An optical model potential describes the e9ective interaction between two particles whose centers of mass are a distance r apart, and for which the solutions of the one-body SchrRodinger equation can be used in an asymptotic sense.) Another signiAcant parameter in nuclear scattering is the Sommerfeld parameter n=
kZ1 Z2 e2 2E
for a collision with center-of-mass energy E between two nuclei with charge numbers Z1 and Z2 : When n is small enough, the Coulomb interaction is not dominant, and there are similarities to scattering from an impenetrable sphere (and its concomitant di9ractive features [186,187]). For larger values of n (as the impact parameter decreases, and neglecting absorption), the changing relationship between the Coulomb repulsion and the nuclear attraction leads to a maximum in the deTection function: a Coulomb rainbow. As the impact parameter further decreases, the ‘battle’ is now between the nuclear attraction and the centrifugal barrier; this time the minimum of the deTection function corresponds to a nuclear rainbow [10]. Not surprisingly, the e9ect of absorption is to damp out contributions from paths positioned below Rs ; uniform rainbow approximations incorporating absorption have been developed [188,189]. A so-called ‘nearside–farside’ decomposition was introduced by Fuller [189] and extensively reviewed more recently by Hussein and McVoy [190]; the Coulomb rainbow is a nearside feature because such trajectories are repulsive, and nuclear rainbows are farside features because the latter paths are attractive. In particular, a nuclear rainbow occurs in the scattering of 5-particles by 90 Zr [190,191]. Fuller and Mo9a [192] extended CAM theory to include the Coulomb interaction. For a surface-transparent potential with weak absorption, the scattering amplitude may be decomposed
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into contributions from barrier waves (a direct reTection term corresponding to reTection at the Coulomb barrier) and internal waves (corresponding to the e9ects of the nuclear potential); see [193,194]. In 1982, Hussein et al. [195] suggested that forward glories (observed of course in optics and atomic scattering) might also occur in heavy-ion nuclear scattering. In optical scattering, there is interference with forward di9raction, but in the nuclear case interference with Coulomb scattering can also arise; this complicates the situation considerably because the latter has a singular amplitude [196] as the forward direction is approached. Under these circumstances, a modiAcation of the total cross section was proposed: it is the sum-of-di;erences cross section
d /C d /el /sod (0 ) ≡ 2 sin d ; − d0 d0 0 representing the cumulative e9ects of the di9erence between the Coulomb and elastic di9erential cross sections, extending down to a small angle 0 : If this angle is small enough, it has been shown that [197,198] 4 0 /sod (0 ) ≈ /R − |fN (0)|sin arg fN (0) − 2/0 + 2n ln sin ; k 2 /R being the total reaction cross section, fN (0) the forward nuclear scattering amplitude, /0 the s-wave Coulomb phase shift and n the Sommerfeld parameter. There are correction terms to this formula which can be ignored if 0 is very small [199 –201]. The above result is the charged-particle analog of the optical theorem /=
4 Im f(k; 0) k
where /sod (0 ) expresses the removal of the Tux from Coulomb paths by absorption and elastic scattering within a narrow forward cone [196]. Note the singular behavior of /sod (0 ) as 0 → 0 as evidenced by the ln sin 0 =2 term: the oscillations are of constant amplitude but increasingly fast frequency. If observed, these could be a characteristic signature of a forward nuclear glory, but in an interesting series of papers [199 –201] (particularly the last paper, from a mathematical standpoint) Ueda et al. argue that forward nuclear glory scattering is not taking place in almost any heavy-ion collisions at low energies, despite the above ‘signature’. They proposed an alternative mechanism, summarized in the almost-poetic title Glory “in the shadow of the rainbow” [201]. This refers to the shadow e9ect of a nuclear rainbow, which leads to a very similar angular distribution to that of the forward glory (being described by the zeroth-order Bessel function, the (Axed) frequency of which is given by the nuclear rainbow angular momentum instead of the glory angular momentum). As far as particle physics is concerned (i.e. high-energy nucleon–nucleon scattering), the concept of tunneling associated with surface waves has been exploited to describe hadronic di;raction [202,203]. In elastic collisions, hadrons (elementary particles that interact via the strong force) can be treated as extended objects in that there is a domain of interaction with a characteristic shape and size (in the center-of-mass system). Typically this is a prolate spheroid, with a constant transverse radius RT (of order 1 fm) and a longitudinal radius RL that is directly
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proportional to the momentum k, i.e. RL ˙ kR2T . Under appropriate circumstances the interaction can then be treated as the scattering of a plane wave by a k-dependent impenetrable prolate spheroid, with the scattering amplitude expressible in terms of a (by now, familiar) background integral plus residue series from Regge-like poles [204,203] (representing surface waves over the spheroid with associated tunneling features). In [203] this model is extended to include an extra parameter to represent blurring of the sharp boundary of the interaction region. We conclude this subsection by noting two papers providing (amongst other features) very useful background information for heavy-ion scattering in particular. Brandan et al. [205] provide a valuable description of semi-classical analysis in the context of optical potential models, and more recently Brandan and Satchler provided a clear review of the physics of ‘light heavy-ion’ scattering [206]. 7. The rainbow as a di'raction catastrophe An alternative way of describing the rainbow phenomenon is by way of catastrophe theory, the rainbow being one of the simplest in catastrophe optics. A review of this subject has been made by Berry and Upstill [208] wherein may be found an introduction to the formalism and methods of catastrophe theory as developed particularly by Thom [209], but also by Arnold [210]. The books by Gilmore [211] (Chapter 13 of which concerns caustics and di9raction catastrophes) and Poston and Stewart [212] are noteworthy in that they also provide many applications. The terminology “di9raction catastrophe” was introduced by Trinkaus and Drepper [213] who were particularly interested in two-dimensional di9raction and the corresponding inverse problem: what information can be obtained from the observed di9raction catastrophe pattern? Connor [214] has utilized the ideas of catastrophe theory to semiclassical collision theory (atomic and molecular). SpeciAcally, has applied catastrophe theory to the study of molecular collisions; in particular, the cusp catastrophe is applied to the theory of cubic or cusped rainbows [215 –217], of interest in atom–molecule collisions [215,216] and the scattering of atoms from surfaces [217–219]. Optics is concerned, to a great degree with families of rays Alling regions of space; the singularities of such ray families are caustics (see Figs. 3, 5 and 24(a),(c)). For optical purposes this level of description is important for classifying caustics using the concept of structural stability: this enables one to classify those caustics whose topology survives perturbation. Structural stability means that if a singularity S1 is produced by a generating function ,1 (see below for an explanation of these terms), and ,1 is perturbed to ,2 ; the correspondingly changed S2 is related to S1 by a di;eomorphism of the control set C (that is by a smooth reversible set of control parameters; a smooth deformation). In the present context this means, in physical terms, that distortions of the raindrop shape of incoming wavefronts from their ‘ideal’ spherical or planar froms does not prevent the formation of rainbows [10], though there may be some changes in the features. Another way of expressing this concept is to describe the ‘system’ as well-posed in the limited sense that small changes in the “input” generate correspondingly small changes in the “output”. For the elementary catastrophes [209], structural stability is a generic (or typical) property of caustics. Each structurally stable caustic has a characteristic di9raction pattern, the wave function of which has an integral representation in terms of the standard
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polynomial describing the corresponding catastrophe. From a mathematical point of view these di9raction catastrophes are especially interesting because they constitute a new hierarchy of functions, distinct from the special functions of analysis [208]. Before summarizing the mathematical details necessary for the purposes of this review, it is appropriate to discuss in qualitative terms the basic rainbow di9raction catastrophe. As known from geometrical optics, the scattering deviation angle D has a minimum corresponding to the rainbow angle (or D7escartes ray) when considered as a function of the angle of incidence i. Clearly the point (i; D(i)) corresponding to this minimum is a singular point (approxi◦ ◦ mately (59 ; 138 )) insofar as it separates a two-ray region (D ¿ Dmin ) from a zero-ray region (D ¡ Dmin ) at this geometrical-optics level of description. This is a singularity or caustic point. The rays form a directional caustic at this point, and this is a fold catastrophe (symbol: A2 ), the simplest example of a catastrophe (see Fig. 24(a); however, it has been noted by McDonald [37] that the caustic is more complicated inside the drop than had previously been realized: the wavefront locus exhibits an odd cusp on a segment of the virtual caustic lying wholly within the raindrop). It is the only stable singularity with codimension one (the dimensionality of the control space (one) minus the dimensionality of the singularity itself, which is zero). In space ◦ the caustic surface is asymptotic to a cone with semiangle 42 (see Fig. 24(c)). Alternatively, as in [208], one can regard i as a state (or behavior) variable and D as a control variable; i(D) is of course multivalued (or not a function, depending on one’s preference); and it seems more causally appropriate to consider D as the state variable. Di9raction is discussed in [208] in terms of the scalar Helmholtz equation ∇2 (R) + A2 N 2 (R) (R) = 0
(7.1)
for the complex scalar wavefunction (R): The concern in catastrophe optics is to study the asymptotic behavior of wave Aelds near caustics in the short-wave limit A → ∞ (semi-classical theory). In a standard manner, (R) is expressed as (R) = a(R)eiA4(R) ;
(7.2)
where the modulus a and the phase A4 are both real quantities. To the lowest order of approximation 4 satisAes the Hamilton–Jacobi equation and can be determined asymptotically in terms of a phase-action exponent (surfaces of constant action are the wavefronts of geometrical optics). The integral representation for is
A n=2
(R) = e−in =4 : : : b(s; R) exp[iA,(s; R)] d n s ; (7.3) 2 where n is the number of state (or behavior) variables s and b is a weight function. In general there is a relationship between this representation and the simple ray approximation (see [208] for further details). According to the principle of stationary phase, the main contributions to the above integral for given R come from the stationary points, i.e. those points si for which the gradient map 9,= 9si vanishes; caustics are singularities of this map, where two or more stationary points coalesce. Because A → ∞, the integrand is a rapidly oscillating function of s so other than near the points si , destructive interference occurs and the corresponding contributions are negligible. The stationary points are well separated provided R is not near a caustic;
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Fig. 24. (a) A generic representation of the fold catastrophe showing the caustic separating the shadow region from the lit or illuminated region. A detailed analysis (see [6]) reveals the underlying nature of the transition zone between the two regions. (ModiAed from [212].) (b) The shadow and lit regions from a spherical raindrop according to geometrical optics. The observer is at O; AB is the axis of symmetry for rays from the sun (S). (c) A schematic representation of the caustic associated with rainbow scattering as observed at O. The caustic (greatly exaggerated ◦ near the raindrops r and r ) is asymptotic to a cone of semi-vertex angle ≈ 42 ; in practice the cone extends to within a few diameters of the drop. Each “color” corresponds to a di9erent (but concentric) cone from each drop, so the observer sees every “color” from a di9erent drop (see Fig. 12:13 in [212]). Rays from near the drop at r in direction BO contribute to the increased illumination below the rainbow seen by the observer at O. The same drop contributes to the rainbow seen by the observer at O . The Agure has axial symmetry about the line PO. It is the existence of the fold caustics that enhance the intensity of scattered light at the rainbow angles (one for each color, though of course there is a continuum rather than a discrete set of angles; in this sense the rainbow has an inAnite spectrum of “colors”). If such caustics were not present, there would be no rainbow, merely an averaging out to white light again.
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the simplest form of stationary phase can then be applied, and yields a series of terms of the form (R ) ≈ aP exp[igP (A; R)] ; (7.4) P
where the details of the gP need not concern us here. Near a caustic, however, two or more of the stationary points are close (in some appropriate sense) and their contributions cannot be separated without a reformulation of the stationary phase principle to accommodate this (i.e. via the CFU method [72]), or by using di9raction catastrophes. The problem is that the “ray” contributions can no longer be considered separately; when the stationary points approach closer than a distance O(A−1=2 ) the contributions are not separated by a region in which destructive interference occurs. When such points coalesce, ,(s; R) is stationary to higher than Arst order, and quadratic terms as well as linear terms in s − sP vanish. This implies the existence of a set of displacements dsi , away from the extrema sP , for which the gradient map 9,= 9si still vanishes, i.e. for which 92 , dsi = 0 : (7.5) 9si 9sj i The condition for this homogeneous system of equations to have a solution (i.e. for the set of control parameters C to lie on a caustic) is that the Hessian 2 9 , =0 ; (7.6) H (,) ≡ det 9si 9sj at points sP (C) where 9,= 9si = 0 (again, details can be found in [208]). The caustic deAned by H = 0 determines the bifurcation set for which at least two stationary points coalesce (in the present circumstance this is just the rainbow angle). In view of this discussion there are two other ways of expressing this: (i) rays coalesce on caustics, and (ii) caustics correspond to singularities of gradient maps. To remedy this problem the function , is replaced by a simpler “normal form” Q with the same stationary-point structure; the resulting di9raction integral is evaluated exactly. This is where the property of structural stability is so important, because if the caustic is structurally stable it must be equivalent to one of the catastrophes (in the di9eomorphic sense described above). The result is a generic di9raction integral which will occur in many di9erent contexts. The basic di9raction catastrophe integrals (one for each catastrophe) may be reduced to the form
1 M(C) = : : : exp[iQ(s; C)] d n s ; (7.7) (2 )n=2 where s represents the state variables and C the control parameters (for the case of the rainbow there is only one of each, so n = 1). These integrals stably represent the wave patterns near caustics. The corank of the catastrophe is equal to n: it is the minimum number of state variables necessary for Q to reproduce the stationary-point structure of ,; the codimension is the dimensionality of the control space minus the dimensionality of the singularity itself. It is
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interesting to note that in “ray” catastrophe optics the state variables s are removed by di9erentiation (the vanishing of the gradient map); in “wave” catastrophe optics they are removed by integration (via the di9raction functions). For future reference we state the functions Q(s; C) for both the fold (A2 ) and the cusp (A3 ) catastrophe; the list for the remaining Ave elementary catastrophes can be found in Table 1 of [208]. For the fold Q(s; C) = 13 s3 + Cs
(7.8)
and for the cusp Q(s; C) = 14 s4 + 12 C2 s2 + C1 s :
(7.9)
As emphasized in [208], the di9raction catastrophes M(C) provide transitional approximations, valid close to the caustic and for short waves. They are increasingly inaccurate far from the caustic, but the theory of uniform approximation (mentioned throughout this review but initially in Section 4 regarding the work of Berry on semi-classical scattering [68]) can be obtained by deforming the standard integrals (see references in [208] for further details). The technique is more general than catastrophe optics, since it can be applied to structurally unstable caustics, the glory being an example, the codimension of which is inAnite [208]. This last statement needs some clariAcation. Because (i) the wavefront associated with the glory possesses circular symmetry, and (ii) there is an inAnite number of topologically di9erent ways in which the (continuous group) symmetry of a circle may be broken, it follows that in order to And all the structurally stable caustics “near” the glory requires inAnitely many parameters [220]; indeed, the singularity index R (see below) for the glory is 12 which means that it is a stronger singularity than the elementary catastrophes with codimensions 1–3 (see the table in Section 3 of [220]). The forward di9raction peak from localized scattering objects is an even stronger structurally unstable caustic than the glory; for further details on short-wave Aelds and Thom’s theorem, again refer to [220]. By substituting the cubic term (7.8) into (7.7) we see that 1 M(C) = √ 2
∞
−∞
√ exp[i(s3 =3 + Cs)] d s = 2 Ai(C) ;
(7.10)
which has been encountered several times before! For C ¡ 0 there are two rays (stationary points of the integrand) whose interference causes oscillations in M(C); for C ¿ 0 there is one (complex) ray that monotonically and exponentially decays to zero. This describes di9raction near a fold caustic. Some 50 years after Airy introduced this function [21] to study di9raction along the asymptote of a caustic (although he did not express it in these terms), and provided a fundamental description of the supernumerary bows, Larmor [221] obtained power laws for the variation of fringe spacing with wavelength of light and the curvature of the caustic. Note that the corresponding integral for the cusp catastrophe is frequently referred to as the Pearcey integral (based on [222]). The scaling properties found by Airy for the intensity near the caustic and by Larmor for the fringe spacing can be deduced by considering, from (7.3) the “physical”
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fold di9raction catastrophe A 1=2 ∞ M(C ) = exp[iA(s3 =3 + C s )] d s ; 2 −∞
(7.11)
where, following [208], s is proportional to distance along the initial wavefront and C is proportional to distance from the fold caustic. Using the scaling As3 = s3 implies that
A1=6 ∞ M(C ) = √ exp[i(s3 =3 + A2=3 C s)] d s : (7.12) 2 −∞ When this is compared with a suggested scaling law between the physical wave function (C ) (dependent on physical controls C , independent of A, e.g. position or time) and the di9raction catastrophe M(C), where C represents the standard control parameters, i.e. in general, (C ) = AR M(A−/i ; Cj ) ;
(7.13)
then it follows that R = 16 and /1 = − 23 . The exponent R is a measure of the divergence of | | as A → ∞ at the most singular control point C = 0 (it is therefore called the singularity index [210]). The /i (there being only one for the rainbow problem) measure the fringe spacings in the di9erent control directions Cj as A → ∞ [223]. Thus near a fold catastrophe the intensity | |2 is O(A1=3 ) [21] and the fringe spacing is O(A−2=3 ) [221]. Note that in contrast to [208] the exponent of A in the argument of M here is negative; it is written in this way without loss of generality. It is noted in [208] that wavelength scaling laws outside catastrophe optics do not necessarily follow the functional form (7.13), an example being the glory (Section 3). As noted in that section, the glory, crudely put, is a result of the intense backscattering of light from spherical √ water droplets. For refractive indices 2 6 N 6 2 (obviously excluding water) rays emerging in the backward direction after one or more internal reTections would form a (structurally unstable) focal line extending to inAnity with intensity | |2 ˙ A. For water a tangentially-incident ray ◦ emerges after one internal reTection at an angle of 14 to the backward direction with amplitude O(A−1=6 ). These surface waves creep around the surface of the drop by di9raction, su9ering wavelength-dependent attenuation as they do so. These rays deAne a focal line, giving rise to a backscattered wave amplitude [6,208] | | ˙ A1=3 exp[ − h(Aa)1=3 ] ;
(7.14)
where a is the drop radius and h is a constant. Note that the glory disappears in the geometricaloptics limit. A very interesting set of papers published by Marston and coworkers [224 –228] take these ideas to the next level: generalized rainbows arising via the cusp di9raction catastrophe for scattering from spheroidal drops (see also [229] for a more classical approach to rainbow scattering). Using acoustically levitated drops whose shapes were closely approximated by oblate spheroids with the short (symmetry) axis vertical. These drops were observed to scatter in the horizontal rainbow region with patterns corresponding to the hyperbolic–umbilic di;raction catastrophe (in Arnold’s classiAcation, D4+ ). An important parameter is the axis ratio q = D=H where D is the diameter in the horizontal plane, and H is in the vertical plane. For q suYciently close to unity, the fold di9raction catastrophe (A2 ) is observed, as expected. As q increases, a
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Fig. 25. (a) The partitioning of angular regions according to the hyperbolic–umbilic di9raction catastrophe, arising in connection with rainbow scattering from spheroidal drops (see [224,230] for details). (b) The rays in the horizontal plane (1,2) and the skew rays (3,4) contribute to the generalization of the rainbow for scattering from spheroidal drops. (Redrawn from [224].)
cusp di9raction catastrophe (A3 ) enters the Aeld for large scattering, and the arcs of the A2 pattern become noticeably bent (see the photographs in [224,225]). For q at or very close to a critical value, found experimentally to be 1:305 ± 0:016, the fold and cusp patterns merge to form a focal section of the D4+ pattern, namely the hyperbolic–umbilic catastrophe. For q beyond this value, the pattern returns to separate A2 and A3 patterns (e.g. for q 1:37 and D = 1:40 mm in [225]). The A3 pattern is described by the Pearcey di9raction integral mentioned above. Why does the D4+ pattern occur? It transpires that for the spheroid there are two once-reTected and twice-refracted rays that are not con?ned to the horizontal plane P, but which merge with the D7escartes ray when 1=2 3N 2 q = qc = 1:311 for N = 1:332 (7.15) 4(N 2 − 1) (see below for discussion of this result), clearly consistent with the above experimental determination. These skew rays are consistent with there being at most four stationary points of the phase function within the D4+ di9raction integral. When q = qc the scattering pattern is divided into three regions corresponding to 0-, 2- and 4-ray regions (see Fig. 25); these reTect the number of non-degenerate points of stationary phase of the integral. Three rays merge for q = qc (the two skew rays and one equatorial ray) in the direction of the cusp point in the focal direction (; >) = (c ; 0), where the Arst and second coordinates refer to the horizontal and vertical scattering angles, respectively, and c is the cusp location at merging. Marston derives a parametric representation of q(c ) which is in good agreement with the experimental data: q = N [2(N 2 − sin2 i)1=2 ((N 2 − sin2 i)1=2 − cos i)]−1=2
(7.16)
and ◦
3 = 180 + 2i − 4 arcsin(sin i=N ) :
(7.17) ◦
His theory predicts that as grazing incidence is approached (i → 90 ) q → qT = N (2N 2 − 2)−1=2 1:070 and
◦
→ T 166 :
This is a Fock-type transition region [5] for the equatorial ray 1 (see Fig. 25); in a spherical ◦ drop the 2-ray region is conAned to the angular interval R 6 6 166 . For qT ¡ q ¡ qc , c
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corresponds to the merging of rays 1; 3 and 4. As i → 0 , 1=2 N q → qL = ; 2N − 2
(7.18)
where qL corresponds to a “lips” event [208,230]. The interval qc ¡ q ¡ qL corresponds to the merging of rays 2– 4. Both types of merging have been photographically conArmed. The scaling laws for the intensity (more accurately, irradiance [22]) I at the focal angles of the A2 ; A3 and D4+ catastrophes are p 2 D D I˙ (7.19) R for Axed q, where p = 13 ; 12 and 23 , respectively, and R is the far Aeld distance. Marston suggests that measurements of c − R may be used for the inverse problem of determining q, even if D is not well known, since c is independent of D . A fascinating prediction is made on the basis of the results in these papers: for a low-altitude sun, oblate-shaped drops might exhibit a “cusped rainbow” (A3 focus) in addition to the usual A2 rainbow. It is suggested that this may occur for D & 1:8 mm. Further details concerning coloration and technical details concerning, for example, the opening rate of the cusp caustic may be found in [226 –228], and references therein. Immediately following (i.e. adjacent to) the paper by Marston and Trinh [224] was one by Nye [230] who provided further details of the catastrophe-theoretic interpretation of their experimental and analytical results. By examining the point of entry of the incident ray a vertical distance j above the horizontal plane, he shows that to Arst order in j the direction cosines of the incident ray, normal to the surface and refracted ray are the following rows of a determinant: sin i 0 cos i 0 ; H=G 1 sin r H=s cos r where s = D cos r and G is the radius of curvature of the surface in the vertical plane, H 2 =2D. For these three directions to be coplanar the determinant must be zero; this condition yields the result 1=2 D 3N 2 qc = = sec r = ; (7.20) H 4(N 2 − 1) which reconArms the analysis of [224]. Nye further predicts the occurrence of a new phenomenon (a lips event, in the terminology of [132]) as N → 2. For further details consult the paper by Nye [230]. When rainbows are viewed in the laboratory at large distances (compared with the size of the drop) from single water droplets they are frequently referred to as colored glarespots [231– 234]. Lock studied their formation using Mie theory and the results from the complex angular momentum theory of Nussenzveig [231]. These caustic surfaces are formed by the light rays
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emerging from the droplet and the focusing properties of the eye. Some of the emerging rays in the lit region form a real caustic beginning near the surface of the drop and ending on the D7escartes ray; the remainder, traced backwards, form a virtual caustic on the rear side of the drop and extending backward with the D7escartes ray as asymptote. These caustics then act as real or virtual sources of the glare spots. In his paper Lock remarks that when an observer is far from a droplet, he sees the Mie scattered intensity, whereas close to the droplet he sees the square of the Fourier transform of the Mie scattered electric Aeld. He suggests that this Fourier-transforming property of the eye may account for the occasional reports of the tertiary rainbow seen in the atmosphere. Noting that the Mie inAnite series of partial waves is an exact solution for the scattering of plane electromagnetic waves from a sphere, Lock points out that all scattering e9ects that can possibly occur are “hidden somewhere or another” in the Mie amplitude. The extraction of this information has been at times problematical, but by Fourier transforming the Mie Aelds, many signals (with their own periodicity) superposed in the spatial domain, become separated in the Fourier domain. This helps to determine the properties of glare spots that are determined by geometrical considerations and those which may not be; in particular a glare spot produced by rays at grazing incidence appears in the Fourier transformed Mie Aelds was noted, the physical mechanism of which was unclear.
8. Summary In summarizing the di9erent aspects of the rainbow discussed in this review, it seems appropriate to identify some categories that, although a little vague and not mutually exclusive, may serve to describe that phenomenon in a variety of di9erent contexts. These are somewhat heuristic in character, but nonetheless may serve as a reminder of the many complementary levels of description that can be appreciated from the viewpoint of mathematical physics. A rainbow is: (1) A concentration of light rays corresponding to a minimum (for the primary bow) of the deviation or scattering angle D(i) as a function of the angle of incidence i; this minimum is identiAed as the D7escartes or rainbow ray. (2) A caustic, separating a 2-ray region from a 0-ray (or shadow) region. (3) An integral superposition of waves over a (locally) cubic wavefront (the Airy approximation). (4) In part, an interference problem (the origin of the supernumerary bows). (5) A coalescence of two real saddle points. (6) (i) A result of scattering by (or interactions with) an e9ective potential comprising a square well and a centrifugal barrier. (6) (ii) An example of Regge-pole dominance. (7) Associated with tunneling in the edge domain (see Section 5.6 and Fig. 13). (8) DeAned, to a considerable extent by the behavior of the scattering amplitude for the third term in the Debye series expansion. (9) A tangentially polarized circular arc. (10) A fold di9raction catastrophe.
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Since much of this review is devoted to an exposition of the complex angular momentum theory of the rainbow (and to a lesser extent the glory), it is of interest also to summarize the di9erences as well as similarities between the predictions of the Airy theory (of particular historical importance) and CAM theory: 8.1. The rainbow according to CAM theory While conArming some aspects of the Airy theory, the method of complex angular momentum predicts di9erences also: six major features of the CAM theory are summarized in [10]: (i) Rainbow enhancement: The scattering amplitudes predicted by geometrical optics (the zeroth-order WKB term) are O( ); CAM theory predicts O( 7=6 ), or a maximum rainbow intensity enhancement of O( 1=3 ), in agreement with the Airy theory. (ii) Rainbow width: The angular width is O(62 ), much narrower than the penumbral angular width of O(6); hence only a fraction of the total scattered intensity “inhabits” a rainbow. (iii) Rainbow polarization: CAM theory predicts that the parallel polarization intensity is about 4% of the corresponding ‘perpendicular’ one; thus there is almost complete dominance of the latter (for which the electric Aeld vector is tangential to the rainbow arc) in agreement with observations [146,147]. While this is basically consistent with the Airy theory, there are deviations from the latter as functions of the angle of observation and the size parameter. (iv) Angular distribution and supernumerary peaks: The Airy theory is valid only for large size parameters and small deviations from R , within the rainbow peak. Corrections to the theory are nevertheless small here, however, for the perpendicular polarization, only becoming signiAcant for the supernumeraries. Large deviations occur for the parallel polarization because of the dominance of a term involving the derivative of the Airy function Ai. Indeed, CAM theory predicts (correctly, see [127]) that the peaks for the parallel polarization are located near the minima for the perpendicular component, and vice-versa (because constructive interference for the latter become destructive for the former, the reTection amplitudes for the parallel polarization change sign after going through zero at the Brewster angle). (v) Uniform approximation: The dominant contribution to the third Debye term in the rainbow region is a uniform asymptotic expansion. On the bright side of the rainbow, this result matches smoothly with the WKB approximation in the 2-ray region. On the dark side, it does so again, this time with the damped complex saddle-point contribution. Put di9erently, di9raction into the shadow side of a rainbow occurs by tunneling. (vi) Higher-order rainbows: The uniform CAM theory can be extended naturally to higherorder bows. For a Debye term of order p, the critical angle of incidence for a rainbow is, (from Eq. (1.3), where p = k + 1) 1=2 2 N −1 ip = arccos : p2 − 1 This angle increases monotonically with p, so that higher-order rainbows arise from incidence within the edge strip. Note also that, since the angular width of the rainbow is O(p 2=3 ), i.e. growing linearly with p for large values of p, the rainbow peak Tattens with increasing p
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values. Interestingly, while the intensity of higher-order rainbows decreases with p by virtue of losses by internal reTection, this becomes less pronounced as p increases because the reTectivity increases as the edge is approached. The deviations from the Airy theory increase with p for both polarizations. In many respects, the rainbow is, in physical terms at least, a simpler phenomenon than the glory. Khare and Nussenzveig [47] list eight di9erent physical e9ects which together or in part can contribute to this second beautiful occurrence. These are: (i) The edge e;ect: This is the e9ect of incident rays in the domain (again, see Section 5.5 and Fig. 13). These rays are both above and below the barrier of the e9ective potential in this domain, and correspond, respectively, to rays that are incident on the drop from within the Fock penumbral region, and rays that reach the surface by tunneling (barrier penetration). (ii) “Orbiting”: This applies to rays that penetrate the drop at angles close to critical; they undergo nearly total internal reTection and consequently are able to make many circumlocutions within the drop with appreciable damping. Mathematically, many higher-order Debye terms contribute to this e9ect. (iii) Axial focusing: The axial symmetry of the problem and the peripheral nature of the leading contributions to the glory give rise to an amplitude enhancement ∼ 1=2 . (iv) Cross-polarization: This occurs because of interference between contributions of comparable magnitude from the electric and magnetic polarization components, leading to additional structure in the glory pattern. (v) Surface waves: These represent one of the two categories of leading contributions to the glory (the other being complex rainbow rays: see (vi) below). These arise from Regge–Debye poles. In addition to strictly surface waves, waves that subsequently take (or have previously taken) additional shortcuts through the drop also must be accounted for. A limiting case is a “glory ray” contribution, an example of which is the p = 24 case for N given by Eq. (6.4), which yields a Fock-type term (see Fig. 21). (vi) Complex rainbow rays: These arise on the shadow side of higher-order rainbows formed near the backward direction (for a description of these higher-order rainbows and their location relative to the observer see [14,132,133]). Mathematically, they are complex saddle points and physically they occur because of the 1=6 rainbow enhancement factor [61] and the Tattening of the rainbow peak for higher-order rainbows. (vii) Geometrical resonances: These are closed (or nearly so) quasi-periodic orbits associated with both the quasi-periodicity of the glory pattern and the narrow resonances or “spikes”. Such orbits play an important role in the semi-classical description of quantization, and the study of resonances and bound states [235]. (viii) Competitive damping: In general, the glory pattern and its polarization are functions of both and , reTecting the interplay between several competing damping e9ects. The outcome of such competition determines whether or not the leading term is electric or magnetic in character, surface-wave or rainbow-type, and the ordering of Debye contributions. The di9erent types of damping have been noted earlier: surface waves are radiation-damped in tangential directions; rainbow terms are damped as complex rays in the shadow of a caustic; and all terms are subject to damping as a result of multiple internal reTections. For ∼ 102 surface-wave e9ects are generally dominant; rainbow e9ects are dominant for
∼ 103 .
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In [47] the authors make an important point in connection with these many di9erent types of contribution (see also [235 –237]): “The glory shows for the Arst time in optics that di9raction e9ects due to complex orbits (surface waves, complex rainbow rays) may be strongly dominant over geometrical optic ray contributions (real orbits). The moral this suggests is that complex extremals of Feynman path integrals may have to be taken into account in more general situations.” Continuing this theme, at the end of his book [10] Nussenzveig asks the question (raised at the beginning of this article): why complex angular momentum? He draws the conclusion that, in part at least, it is because of the intimate relationship that exists between tunneling and analytic continuation, noting that all descriptions of the former make use of the latter (in one form or another). It is perhaps surprising, initially, that such ‘physical’ and ‘mathematical’ topics should be so closely related, but a moment’s reTection on both the historical and current relationship between the two subjects indicates that this is fascinating but very natural. Indeed, to quote from [10]: “Ultimately, this [relationship] goes back to Euler’s great discovery of the connection between oscillations and the exponential function, exempliAed by what has been called one of the most beautiful formulas of mathematics, his synthesis of analysis, algebra, geometry and arithmetic: ei = − 1.” It is Atting that “one of the most beautiful formulas of mathematics” should provide the basis for a richer understanding of two of the most beautiful meteorological phenomena known. Nussenzveig explains further why CAM theory is especially well suited to the treatment of semiclassical problems: “angular momentum is conjugate to the scattering angle, and the localization principle provides a natural physical interpretation of the Poisson representation, linking Huygens’ principle with pseudoclassical paths. This allows one to combine physical insights derived from classical wave theory and from quantum potential scattering in order to deal with previously intractable dynamical features of di9raction. Thus, the beautiful mathematical theory of analytical continuation provides the key to a deeper understanding of some of the most beautiful phenomena displayed in the sky, and also manifested in so many other ways—through all scales of size—revealing the underlying unity of nature.” It is quite remarkable that, despite the tremendous development of mathematical physics since the time of Newton, he was able to express similar thoughts almost 300 years earlier when he stated in ‘Question 31’ of “Opticks” that “Nature is very consonant and conformable to herself.” [241] Acknowledgements It is a pleasure to acknowledge the assistance of several individuals during the preparation of this manuscript. In particular, I am grateful to Professor H.M. Nussenzveig for permission
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to use some Agures from some of his publications; I have found the book [10] to be a valuable resource, especially with regard to what one might describe as ‘macroscopic tunneling’, but also as a source of references to rainbows and glories in atomic, molecular and nuclear scattering. I am also grateful to Professor M.S. Hussein who provided me with references to some of his most recent publications. Both the anonymous reviewer and the editor, Professor J. Eichler provided valuable feedback and advice which helped to improve the manuscript considerably. In addition, Professor T.F. Gallagher o9ered advice in the early stages of preparation of this article. My colleague Professor Mark Lesley drew my attention to the statement by Newton (quoted above); I am delighted that his enthusiasm for this project has been almost as great as mine. Finally I would like to thank all the sta9 at the editorial oYce in Amsterdam, particularly Yvonne van Lieshout who patiently encouraged me through the entire editorial process.
Appendix A. Classical scattering; the scattering cross section Consider the problem of a particle of mass m moving under the inTuence of a central Aeld; the path of the particle lies in one plane and its angular momentum M is conserved [238,239], i.e. in standard notation, M = mr 2 ˙ = constant :
(A.1)
It is well-known that this implies Kepler’s second law, namely that the radius vector of the particle sweeps out equal areas in equal times. If V (r) is the potential Aeld in which the particle moves, then the total energy E of the particle may be written as 2
E = 12 m(r˙2 + r 2 ˙ ) + V (r) = 12 mr˙2 +
M2 + V (r) ; 2mr 2
from which $ % 2 % 2 M & : r˙ = [E − V (r)] − m mr This expression can be integrated to yield the time t explicitly, or from (A.1)
M = d r + constant ; r 2 {2m[E − V (r)] − (M=r)2
(A.2)
(A.3)
(A.4)
which is the general solution of the equation of the path (r). It also follows from (A.1) that varies monotonically with time since ˙ never changes sign. The quantity M2 + V (r) = U (r) 2mr 2
(A.5)
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Fig. 26. (a) The distribution of scattered particles incident from a narrow annulus of width ]b, b being the impact parameter, located on the surface of a large sphere with center at the scattering center. (b) Two orbits for a repulsive potential with slightly di9erent values of the impact parameter b.
Fig. 27. Relationships between the impact parameter b, the classical distance of closest approach (or outermost radial turning point) r0 , the deTection angle @ and the scattering angle : In (a) the interaction is attractive, and @ = ; in (b) it is repulsive, and @ = − 2 − . (Redrawn from [10].)
is called the “e9ective potential”, being the sum of the central potential V (r) and the “centrifugal” potential. Values of energy for which E = U (r) are intimately related to the limits of the motion since at these values r˙ = 0, thus deAning the turning points of the path. The radial motion of the particle ranges from r = ∞ to the largest root r0 of r˙ = 0 (thus deAning the classical distance of closest approach (or outermost radial turning point), and then from r0 to r = ∞. The total (classical) deTection angle (see Figs. 26 and 27) is therefore
∞ dr : (A.6) @(M ) = − 2M r0 r 2 {2m[E − V (r)] − (M=r)2
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The impact parameter b is deAned as b = M=mv0 , where v0 is the initial speed of the particle, so @ = @(b) also. For an entirely repulsive potential, @ ∈ [0; ], but as pointed out in [10], it can take arbitrarily large negative values for an attractive one, since the particle may “circle” around the scattering center many times before Anally emerging. The relationship between the deTection angle @ and the scattering angle is @ = ± − 2n ;
n = 0; 1; 2; : : :
(see Fig. 27) where is chosen such that ∈ [0; ], its physical range of variation. For particles with impact parameters in the range [b; b + ]b] the area of the associated annuls is ]/ = 2 b|]b|; if these particles are scattered within an “onion ring” of area ]S = 2 R2 sin |]| (see Fig. 26), or equivalently within a solid angle ]0 = 2 sin |]|, then the di;erential scattering cross section is deAned to be (in the appropriate limiting sense) d/ b csc b csc = : = d 0 |d = d b| |d @= d b|
(A.7)
The distribution is independent of the azimuthal angle , for a central force Aeld. Note that for a repulsive potential inparticular, d = d b ¡ 0 because the larger the impact parameter the smaller will be the angle through which the particles are scattered in general. Note that the total scattering cross section is obtained by integrating (A.7) over all solid angles, i.e.
bmax db d/ /= d0 = 2 b d = b2max ; (A.8) d0 d 0 where bmax is the maximal impact parameter, i.e. if for r ¿ bmax the potential vanishes (as it does for square well or barrier potentials) then particles with an impact parameter greater than bmax are not deTected at all (at least, classically; tunneling does occur as pointed out in the body of the article). The scattering cross section is then just the geometrical cross section of the region; if the scattering potential has no such Anite cut-o9 then the classical scattering cross section is inAnite. This is related to a sharp forward peaking of the di9erential cross section [239, Chapter 5], and is one of the classical singularities in d /= d 0 discussed below. It is pointed out in [10] that there may exist several trajectories which give rise to the same scattering angle, so (A.7) must be generalized to sum over all the impact parameters that lead to the same angle: bj ()csc d/ (A.7a) () = d0 |d = d bj | j where the derivative on the right-hand side is equivalent to the Jacobian of the transformation relating and b. Some of these trajectories are illustrated in Fig. 28 for a typical potential type found in atomic and nuclear scattering problems, namely an attractive outer region with a repulsive central core (also shown schematically in the Agure, along with a typical deTection function @(b)). It is also noted in [10] (see also [200]) that there are three di9erent types of singularities (or caustics—a caustic is an envelope of a family of rays) associated with directions in which the di9erential cross section diverges. These are as follows:
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Fig. 28. (a) Several di9erent paths for particle interaction with a potential composed of a repulsive inner core (hatched region) with an attractive outer shell. (b) The corresponding deTection angle as a function of the impact parameter b; @(b): The inset shows the typical potential shape V (r): Note that the repulsive path 2 and the attractive paths 4 and 6 result in the same magnitude of the scattering angle ||. Path 3 (when b = bG ) is a ‘glory path’ and path 5 (when b = bR ) is a ‘rainbow path’. (Redrawn from [10].)
(I) Rainbow scattering: This is a caustic arising when = R , the rainbow angle; it occurs when d =0 ; (A.9) d b R so that the deTection function passes through an extremum. Its name arises from the analogy with the optical rainbow. In the “heavy-ion” scattering this caustic is referred to as Coulomb or nuclear rainbow scattering if the extremum is a maximum or minimum, respectively [200], and is manifested in terms of an angular distribution given by the Airy function (see Section 2.1). Clearly in view of (A.9) for suYciently close to R we may write (b) ≈ R + 12 (bR )(b − bR )2 ; where b = bR when = R . Under these circumstances, the classical di9erential scattering cross section is d/ b 2 ≈ d0 sin R | (bR )( − R )| which diverges like | − R |−1=2 . (II) Glory scattering: This is a caustic caused by the vanishing of sin in Eq. (A.7). In terms of the deTection angle @ for non-zero impact parameter bG @(bG ) = n
(n = 0; −1; −2; : : :)
so that the di9erential cross section diverges like (sin )−1 in the precisely forward direction (n even: forward glory scattering) or backward direction (n odd: backward glory scattering).
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Eq. (A.6) implies that @ = can only occur when b = 0; this case does not constitute glory scattering. The characteristic angular distribution behaves like a zeroth-order Bessel function [200], where it is pointed out that nuclear rainbow scattering and backward glory scattering are expected to be good probes of nuclear interaction at relatively short distances, as is forward glory scattering at relatively large distances. (III) Forward peaking: As alluded to above, scattering potentials with tails extending asymptotically to zero at inAnity will lead in general to a divergent forward di9erential cross section. This is because of the accumulation of small deTections for particles with arbitrarily large impact parameters, i.e. from (A.7a) in which some bj → ∞; → 0 while |d = d bj | remains bounded. Even for cut-o9 potentials, some forward anomaly is still to be expected. There is a fourth situation that may occur when the e9ective potential U (r) = UM (r) for a given value of M has a relative maximum at r = r0 : (IV) Orbiting: In this case UM (r) = E;
(d UM = d r)r0 = 0 :
This is interpreted as the existence of an unstable circular orbit with radius r0 [10]. Under these circumstances the integral (A.6) diverges logarithmically at its lower limit if UM (r0 ) = 0. A simple calculation yields the result that @(M ) → −∞ as r → r0 ; this means that a particle with this energy and angular momentum will spiral indeAnitely around the scattering center, and taking an inAnitely long time to reach the top of the “barrier”. In addition, the inverse function b() becomes inAnitely many-valued, so an inAnite number of branches contribute to the di9erential cross section (A.7a). If UM (r) has a local minimum for r ¡ r0 then particles with energy E ¡ UM (r0 ) can sustain oscillatory orbits within the potential “well”, and the unstable circular orbit functions as a separatrix between bounded and unbounded trajectories [10] (see also [240]). Fig. 29 illustrates connections between the e9ective potential, tunneling and the ray picture in rainbow and glory formation. A.1. Semi-classical considerations: a prBecis In a sense, the semi-classical approach is the “geometric mean” between classical and quantum mechanical descriptions of phenomena; while one wishes to retain the concept of particle trajectories and their individual contributions, there is nevertheless an associated de Broglie wavelength for each particle, so that interference and di9raction e9ects enter the picture. The latter do so via the transition from geometrical optics to wave optics. The di9erential scattering cross section is related to the quantum scattering amplitude f(k; ) and this in turn is expressible as the familiar partial wave expansion [65]. The formal relationship between this and the classical di9erential cross section is established using the WKB approximation and the principle of stationary phase is used to evaluate asymptotically a certain phase integral (see [10, Section 1.2] for further details). A point of stationary phase can be identiAed with a classical trajectory, but if more than one such point is present (provided they are well separated and of the Arst order) the corresponding expression for |f(k; )|2 will contain interference terms. This is a distinguishing feature of the ‘primitive’ semi-classical formulation, and has signiAcant implications for e9ects (I) – (IV) noted above. The inAnite intensities (incorrectly) predicted
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Fig. 29. (a) The e9ective potential U (r; ) for a transparent sphere of radius a (similar to Fig. 13 but with U (r; ) ¿ 0 for all r and redrawn to show four ‘energy levels’, respectively, above the top of the potential well, at the top, in the middle and at the bottom of the well. (b) The corresponding incident rays and impact parameters. Case 2 shows a tangentially incident ray; note that in case 1 the refracted ray is shown. It passes the center at a distance of l = b=N ; that this is the case is readily shown from simple geometry: from Snell’s law of refraction, sin i = N sin r = b=a and since l = a sin r, the result follows. (c) Similar to (a), but with resonant wave functions shown, corresponding to ‘family numbers’ n = 0 and 1 (the latter possessing a single node). (d) The ‘tunneling’ phenomenon illustrated for an impact parameter b ¿ a; being multiply reTected after tunneling, between the surface r = a and the caustic surface r = b=N (the inner turning point). ((a) and (b) redrawn from [10]; (c) and (d) from [88].)
by geometrical optics at focal points, lines and caustics in general are “breeding grounds” for di9raction e9ects, as are light=shadow boundaries for which geometrical optics predicts Anite discontinuities in intensity. Such e9ects are most signiAcant when the wavelength is comparable with (or larger than) the typical length scale for variation of the physical property of interest (e.g. size of the scattering object). Thus a scattering object with a “sharp” boundary (relative to one wavelength) can give rise to di;ractive scattering phenomena. Under circumstances like those in I–IV above, the primitive semiclassical approximation breaks down, and di9raction e9ects cannot be ignored. Although the angular ranges in which such critical e9ects become signiAcant get narrower as the wavelength decreases, the di9erential cross section can oscillate very rapidly and become very large within these ranges. As such the latter are associated with very prominent features and in principle represent important probes of the potential, especially at small distances. The important paper by Ford and Wheeler [19] contained transitional asymptotic approximations to the scattering amplitude in these ‘critical’ angular domains, but they have very narrow domains of validity, and do not match smoothly
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with neighboring ‘non-critical’ angular domains. It is therefore of considerable importance to seek uniform asymptotic approximations that by deAnition do not su9er from these failings. Fortunately, the problem of plane wave scattering by a homogeneous sphere exhibits all of the critical scattering e9ects (and it can be solved exactly, in principle), and is therefore an ideal laboratory in which to test the accuracy and eYcacy of the various approximations. Furthermore, it has relevance to both quantum mechanics (as a square well or barrier problem) and optics (Mie scattering); indeed it also serves as a model for the scattering of acoustic and elastic waves, and was studied in the early twentieth century as a model for the di9raction of radio waves around the surface of the earth [25] (and see Appendix C). Appendix B. Airy functions and Fock functions There are several di9erent notational forms for Airy functions in the literature. In the form introduced by Airy [21], the ‘rainbow integral’ was deAned as [10,242]
31=3 ∞ Ai(z) = cos(+3 + 31=3 z+) d + ; (B.1) 0 where the above form y = Ai(z) is a particular solution (for real z) satisfying the “wound healing” di9erential equation d2 y + zy = 0 d z2
(B.2)
(so-called because it provides a transition between discontinuous potentials or other spatial features). A second linearly independent solution is denoted by Bi(z). The Airy functions Ai(z) and Bi(z) are entire transcendental functions of z and are real when z is real. They may be expressed as linear combinations of Bessel functions of order ± 13 ; either in terms of modiAed Bessel functions (for argument z) or ordinary Bessel functions (for argument −z). Their derivatives are correspondingly expressible in terms of Bessel functions of order ± 23 (Bowman et al.). The more general form of (B.1) for complex values of z is (see also (B.6) below), together with a rather more complicated expression for Bi(z).
∞ei =3 1 1 3 Ai(z) = exp + − z+ d + : 2 i ∞e−i =3 3 The general solution of (B.2) may be written as y(z) = z 1=2 [c1 J1=3 (>) + c2 Y1=3 (>)] ;
(B.3)
where > = 23 z 3=2 : The asymptotic expansion of Ai(z) for large |z | is [243] Ai(z) ≈ 12 −1=2 z −1=4 e−> [1 + O(>−1 )]
(B.4)
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for |arg z | ¡ , and
' ( (4) Ai(−z) ≈ −1=2 z −1=4 sin > + [1 + O(>−1 )] − cos > + O(>−1 ) : 4 4 An alternative form for the Airy integral, used by van de Hulst [7], is
∞ f(z) ˜ = cos (z+ ˜ − +3 ) d + : 2 0
Jackson [35] uses the form [243] 3
1 ∞ + Ai(−z) = cos − z+ d + 0 3
(B.5)
(B.6)
where z = (2k 2 a2 = (0 ))1=3 ( − 0 ) (where z˜ = (12= 2 )1=3 z; and f(z) ˜ = (2 2 =3)1=3 Ai(−z)). The Fock functions are deAned as [244]
e−i =6 eiz+ (5) f(z) = d+ 2 F Ai(+e2i =3 )
eiz+ ei =6 d+ ; g(z) = − 2 F Ai (+e2i =3 )
(B.7)
where F is a contour starting at inAnity in the angular sector =3 ¡ arg + ¡ ; passing between the origin and the pole of the integrand nearest the origin, ending at inAnity in the angular sector − =3 ¡ arg + ¡ =3. They can be generalized to include integer powers of + in the integrand. To leading order, their asymptotic behavior is [244] f(z) ∼ 2iz exp(−iz 3 =3); ∼ 0;
z → −∞
z→∞
and g(z) ∼ 2 exp(−iz 3 =3); ∼ 0;
z → −∞
z→∞:
Appendix C. The Watson transform and its modi.cation for the CAM method The Watson transformation was originally motivated by the desire to understand di9raction of radio waves around the Earth (prior to the discovery of the ionosphere). The topic of primary interest was therefore the neighborhood of the Earth’s surface in the shadow zone of the transmitter. Once the transformation to the complex l-plane had been made, with the
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corresponding freedom of path deformation, rapidly convergent asymptotic estimates could be made by paths suitably chosen to emphasize a few dominant pole contributions as opposed to dealing with those from many partial waves. In the shadow region the dominant (complex) poles n are now called Regge poles, and their imaginary part (associated with angular damping [47]) grows rapidly with n, so as we have seen, only those few poles closest to the real axis need to be considered. This leads to a rapidly convergent residue series, each term of which corresponds to a “creeping wave” generated by tangentially incident rays which are radiation damped as they travel around the surface of the sphere. In the lit region Watson’s method fails: in addition to Regge poles the now-dominant “background integral” in the l-plane (or equivalently the -plane, where = l + 12 ) must be considered. The signiAcant contributions to this integral come from saddle points which are, in general, complex. When these are real, however, they correspond to the rays of geometrical optics and provide the associated contributions; by taking account of higher-order terms in the saddle-point method, the WKB series can be recovered to any desired accuracy. There is a penumbral region between the lit and shadow regions: this was investigated by Fock [245], who was able to describe the behavior of the wave Aeld in this region in terms of a new function (subsequently named the Fock function; see Appendix B). It is in this region, near the edge, that the creeping waves are generated. The Fock function interpolates smoothly between the geometrically lit (WKB) region and the di9racted ray or creeping wave region. While subsequent attempts were made to apply the method to both impenetrable and transparent spheres ([30,31]; see [47] for further references), and signiAcant advances were made, there was no theory going beyond the classic Airy approximation in a comprehensive manner for all scattering angles. In a very comprehensive treatment of wave and particle scattering [239], Newton mentions many of the topics contained in this review (see for example, Chapters 3, 5, 13 and 18 therein). In particular, the latter chapter summarizes the CAM method as originally formulated by Regge, and the reader is referred there for details of the Watson transform (unmodiAed), questions of uniqueness, Regge poles and the Mandelstam representation. In this appendix an outline due to Frautschi [246] is followed for the Watson transform, and further commentary on the modiAed transform is added from [47]. There are many equivalent but seemingly di9erent representations of the Watson–Regge transform (as it is called in [239]; see for example [247], Chapter 15 and references in [248]); a common one will be adopted here. We consider the partial-wave expansion of the scattering amplitude in form (5.18) (omitting the drop diameter or well radius a): f(k; ) =
∞
1 (2l + 1)(Sl (k) − 1)Pl (cos ) 2ik
(C.1)
l=0
and consider the analytic continuation of this expression to the complex l-plane, Re l ¿ − 12 : This partial-wave expansion may be rewritten as an integral I(k; ) in this domain such that each term in the summation is the residue of a pole in the contour integration; thus
P (−cos ) 1 I(k; ) = (2l + 1)(Sl (k) − 1) l dl ; (C.2) 4k C sin l
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Fig. 30. The (classical) Watson transformation contour for the scattering amplitude before (P) and after deformation away from the real l-axis. A corresponding contour applies in the complex -plane also.
where C is the contour shown in Fig. 30. This representation follows from the fact that near the pole of the integrand at l = n ¿ 0; Pn (−cos ) Pn (cos ) Pl (−cos ) ≈ = n sin l (−1) (l − n) (l − n) and the orientation of the contour C , which encloses all the integers l = 0–N, and the limit N → ∞ is taken. The original expansion (C.1) is not useful for z = cos ∞ → ∞ because it converges only at small z, i.e. within the Lehmann ellipse [239,246 –248], the diYculty lying with the asymptotic behavior of Pl (z). Regge overcame this problem [249] by shifting the contour of integration from around the real l-axis to the vertical line l = − 12 + j + i Im l; this is satisfactory provided the contribution from the semicircular contour vanishes as l → ∞ and any additional singularities accrued by the shift in contour are suYciently well-behaved. Regge investigated this representation for superpositions of Yukawa potentials at large values of l; he found that the number of poles for Re l ¿ N is Anite, and that the semi-circular contribution vanished at large l for small z subject to certain technical details (see also [250 –252]). Then I(k; ) can be written as a sum over poles plus a “background integral” along Re l = − 12 + j:
i (k) 1 i∞−1=2+j P (−cos ) I(k; ) = (2l + 1)(Sl (k) − 1) l dl + P5 (k) (−cos ) 4k −i∞−1=2+j sin l sin 5i (k) i i (C.3)
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in terms of the pole contributions 5i (k) with factor i . This is sometimes called the Sommerfeld– Watson representation, and now the pole furthest to the right (51 say, with Re l ¿ − 12 ) dominates the integral, i.e. as z = cos → ∞, I(k; z) ∼
1 (k) 51 z : sin 51 (k)
(C.4)
In general, as noted in Section 5, for potentials which are not superpositions of Yukawas (e.g. square well or other Anite-range potentials), the contribution from the semicircular contour does not vanish (but see [250]). The Yukawa potentials thus are of great importance for the Mandelstam representation [92] (which cannot exist unless I is bounded by a Anite power of z; (C.4) is also of signiAcance in connection with relativistic scattering [246]). To elucidate part of the physical signiAcance of (C.3), consider the inner product
1 1 Il (k) = P (z)I(k; z) d z ; 2 −1 l along with the relation
1 1 sin 5 P (z)P5 (−z) d z = 2 −1 l (5 − l)(5 + l + 1)
(C.5)
where l is an integer and 5 is complex [243]. (This reduces to the standard result (−1)l =(2l+1) as 5 → l). Consider next the contribution of one Regge pole at l = 5 to Il , namely
(k) : (5(k) − l)(5 + l + 1)
(C.6)
If 5 ≈ l, this can provide a large contribution which may change rapidly with k (and hence energy), so if there exists an energy El such that l = Re 5, then expanding 5 in a Taylor series about El yields 5 l + (E − El )
d Re 5(El ) + i Im 5(El ) : dE
(C.7)
Then (B.6) can be rewritten in Breit–Wigner form as
(5 + l + 1)[d Re 5(El )= d E]{E − El + iF=2}
(C.8)
where Im 5(El ) F = ; 2 d Re 5(El )= d E
I is the resonance half-width. Interpretation of these results to trace the trajectory of the Regge pole may be found in [246,249]. As noted in Section 5, a detailed analysis of a more e9ective version of the Watson transformation appeared in 1965 [33], along with many subsequent applications (see Section 5 for details). This modi?ed Watson transformation is based on the application of the Poisson sum
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formula [253,254] to the scattering amplitude or the wave Aeld [33]; one form of this formula is:
∞ ∞ ∞ 1 m $ l + ; ; = (−1) $( ; ; ) exp(2im ) d (C.9) 2 0 m=−∞ l=0
with an appropriate choice of deformed path in the complex -plane. This choice, which is di9erent for di9erent angular regions, leads also to Regge pole contributions and saddle-pointdominated background integrals. Khare and Nussenzveig point out that a saddle-point on the real axis is also a point of stationary phase in (C.9), and this characterizes an extremal path, i.e. a ray in geometrical optics, or a classical orbit in the particle context [47]. The integer m in (C.9) has the topological signiAcance of a winding number, which is associated with the number of circumlocutions performed by a path around the center of the sphere. The details and subtleties of the Regge poles for the problem of scattering by a transparent sphere, the basis of the rainbow and the glory, are discussed in detail in Section 5.2, along with the signiAcance of the Debye expansion in regaining rapid convergence of the residue series. A succinct derivation of the Poisson sum formula has been provided in the book by Brink [186], and in view of its importance in CAM theory in particular, the formal proof is presented here (in a slightly modiAed form). From the theory of (complex) Fourier series, with standard Dirichlet conditions applying to f(x) [255], it follows that ∞ f(x) = Am e−2m ix=L ; m=−∞
where
1 b+L Am = f( )e2m i =L d : L b Note that since m and if l are integers, ∞ f(x + l) = Am e−2m ix=L e−2m ix=L m=−∞
then choosing L = 1 it follows that ∞ Am e−2m ix ; f(x + l) = m=−∞
where
Am =
l
l+1
f( )e2m i d ;
having chosen b = l. Therefore in particular ∞ ∞ 1 (6) f l + = Am e−m i = (−1)m Am 2 m=−∞ m=−∞ =
∞ m=−∞
m
(−1)
l
l+1
f( )e2m i d :
(C.10)
J.A. Adam / Physics Reports 356 (2002) 229–365
Hence ∞ l=0
359
∞ ∞ ∞ ∞ l+1 1 m 2m i m f l+ = (−1) f( )e d = (−1) f( )e2m i d 2 l 0 m=−∞ m=−∞ l=0
(C.11)
and all that remains is to replace f( ) by $( ; ; ) to recover (C.9). Appendix D. The Chester–Friedman–Ursell (CFU) method This technique is used when two saddle points in the complex -plane approach one another as a parameter governing the conTuence gradually changes. The description given in [10] is followed here, notationally adapted to the rainbow problem. The complex integral of interest is
˜ ˜ ˜ ˜ F( ; ) = g( )e f( ; ) d ; (D.1) where ˜ = 2 , the asymptotic expansion parameter is large and positive and ˜ = − R is an independent parameter. In the 1-ray region (in particular), the integral is dominated by a single ` ); ˜ in the neighborhood of which f and g are regular, so saddle-point ` = ( ` 2: ` ) ˜ + 1 f ( ; ` )( ˜ ˜ ≈ f( ; − ) f( ; ) 2 ` a Gaussian-type integral occurs, When the path of steepest descent is chosen to pass through , ˜ ) ˜ in a range ] about the saddle and this yields the dominant contribution to the integral F( ; point, where [103] ` ) ˜ |)−1=2 : ] ∼ ( ˜ |f ( ; ˜ ) ˜ then follows by standard techniques (a power-series The asymptotic expansion of F( ; ` expansion of g about and subsequent integration term by term [104]). By contrast, in the 2-ray region there are two saddle points ( ` ; ` say) to be considered, and provided their ranges do not overlap, their contributions are additive. When the ranges do overlap, the above method is inapplicable because the series expansion of g has a radius of convergence of order | ` − ` |: Furthermore, at conTuence (when ˜ = 0), the second derivative f is zero, so the second term in f above becomes cubic in . For a decreasing sequence of ?xed ˜ = 0; ˜ can be chosen large enough so that the ranges still do not overlap (i.e. ] → 0 as ˜ Similarly, a cubic approximation
˜ → ∞), but this does not yield a result uniformly valid in . ˜ ˜ ˜ ): ˜ for f( ; ) about = 0 provides only a transitional approximation for F( ; The basis of the CFU method is to transform the exponent in the equation (D.1) into an exact cubic by means of the change of variables ˜ = 1 u3 − +()u ˜ + A() ˜ f( ; ) 3 ˜ The essential feature is to design a mapwith the two saddle points transformed into ±+1=2 (): ping which preserves the saddle point structure [235]. Under these circumstances the mapping
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↔ u is analytic and one-to-one near u = 0: It follows eventually from (D.1) that an asymptotic expansion of the following form is valid: M −1 −1=3 −s −M 2=3 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ F( ; ) = exp[ A()]
as () + O( ) Ai[ ˜ +()] + ˜
M −1 −2=3
s=0
˜ ˜ bs ()
−s
+ O( ˜
−M
) Ai [ ˜
2=3
˜ +()]
:
s=0
The as and bs are the so-called ‘CFU coeYcients’ determined from the expansion of the integrand after the transformation. When |z |1 and |arg z | ¡ , it is known that [243] Ai (z) ≈ −z 1=2 Ai(z)
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Physics Reports 356 (2002) 367–474
Predictability: a way to characterize complexity G. Bo#ettaa; b;∗ , M. Cencinic; d , M. Falcionid; e , A. Vulpianid; e a
Dipartimento di Fisica Generale, Universita di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy b Istituto Nazionale Fisica della Materia, Unita dell’Universita di Torino, Italy c Max-Planck-Institut f'ur Physik komplexer Systeme, N'othnitzer Str. 38, 01187 Dresden, Germany d Dipartimento di Fisica, Universita di Roma “la Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy e Istituto Nazionale Fisica della Materia, Unita di Roma 1, Italy Received January 2001; editor: I: Procaccia This review is dedicated to our masters and to our friends; in particular to Andrei N. Kolmogorov (1903–1987) and Giovanni Paladin (1958–1996) whose in8uence on our work runs deeper than we can know.
Contents 1. Introduction 2. Two points of view 2.1. Dynamical systems approach 2.2. Information theory approach 2.3. Algorithmic complexity and Lyapunov exponent 3. Limits of the Lyapunov exponent for predictability 3.1. Characterization of
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4.5. Convective chaos and spatial complexity 4.6. Space–time evolution of localized perturbations 4.7. Macroscopic chaos in globally coupled maps 4.8. Predictability in presence of coherent structures 5. Predictability in fully developed turbulence 5.1. Basic facts of turbulence 5.2. Reduced model of turbulence 5.3. E#ects of intermittency on predictability of in
∗
Corresponding author. E-mail address: bo#
[email protected] (G. Bo#etta). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 2 5 - 4
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7. Irregular behavior in discrete dynamical systems 7.1. Dependence of the period on the number of the states 7.2. The transition from discrete to continuous states 7.3. Entropies and Lyapunov exponents in cellular automata 8. The characterization of the complexity and system modeling 8.1. How random is a random number generator? 8.2. High-dimensional systems 8.3. Di#usion in deterministic systems and Brownian motion 8.4. On the distinction between chaos and noise
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9. Concluding remarks Note added in proof Acknowledgements Appendix A. On the computation of the
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Abstract Di#erent aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov–Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a characterization of the unpredictability of a system gives a measure of its complexity. Adopting this point of view, we review some developments in the characterization of the predictability of systems showing di#erent kinds of complexity: from low-dimensional systems to high-dimensional ones with spatio-temporal chaos and to fully developed turbulence. A special attention is devoted to
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All the simple systems are simple in the same way, each complex system has its own complexity (freely inspired by Anna Karenina by Lev N. Tolstoy) 1. Introduction The ability to predict the future state of a system, given the present one, stands at the foundations of scienti
In this review we shall always consider the usual setting where a system is studied by an external observer. In this way one can avoid the problem of the self-prediction [192].
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In this review we will discuss how these points of view are related and how they complete each other in giving a quantitative understanding of complexity arising in dynamical systems. In particular, we shall consider the extension of this approach, nowadays well established in the context of low dimensional systems and for asymptotic regimes, to high dimensional systems with attention to situations far from asymptotic (i.e.
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2. Two points of view 2.1. Dynamical systems approach Two standard—tightly linked—indicators are largely used to quantify the behavior of a dynamical system with respect to the asymptotic evolution of an in
(2.2) x ∈ Rd
In both cases, for simplicity, we suppose that a vector uniquely speci<es one state of the system. We also assume that F and G are di#erentiable functions, that the evolution is well-de
x(t)
or, in the case of discrete time maps: d 9Gi zj (t) : zi (t + 1) = 9xj j=1
x(t)
(2.4)
Under rather general hypothesis, Oseledec [169] proved that for almost all initial conditions x(0), there exists an orthonormal basis {ei } in the tangent space such that, for large times, d ci ei ei t ; (2.5) z(t) = i=1
where the coeJcients {ci } depends on z(0). The exponents 1 ¿ 2 ¿ · · · ¿ d are called characteristic Lyapunov exponents. If the dynamical system has an ergodic invariant measure, the
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spectrum of LEs {i } does not depend on the initial condition, except for a set of measure zero with respect to the natural invariant measure. Loosely speaking, (2.5) tells us that in the phase space, where the motion evolves, a d-dimensional sphere of small radius j centered in x(0) is deformed with time into an ellipsoid of semi-axes ji (t) = j exp(i t), directed along the ei vectors. Furthermore, for a generic small perturbation x(0), the distance between a trajectory and the perturbed one behaves as |x(t)| ∼ |x(0)| e1 t [1 + O(e−(1 −2 )t )] :
(2.6)
If 1 ¿ 0 we have a rapid (exponential) ampli
(2.11)
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Let us stress that the Lyapunov exponents give information on the typical behaviors along a generic trajectory, followed for in
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of the contracting directions (associated with the negative Lyapunov exponents) is completely lost. We can experience only the e#ect of the expanding directions, associated with the positive Lyapunov exponents. As a consequence, in the typical case, the coarse grained volume behaves as V (j; t) ∼ V0 e(
i ¿0
i )t
;
when V0 is small enough. Since Ne# (j; t) ˙ V (j; t)=V0 , one has i : hKS =
(2.14) (2.15)
i ¿0
This argument can be made more rigorous with a proper mathematical de
Because of its relation with the Lyapunov exponents—or by the de
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nevertheless in a few years it became an important branch of both pure and applied probability theory with strong relations with other <elds as computer science, cryptography, biology and physics [230]. For the sake of self-consistency we brie8y recall the basic concepts and ideas about the Shannon entropy. Consider a source that can output m di#erent symbols; denote by s(t) the symbol emitted by the source at time t and with P(CN ) the probability that a given word CN = (s(1); s(2); : : : ; s(N )), of length N , is emitted: P(CN ) = P(s(1); s(2); : : : ; s(N )) :
(2.17)
We assume that the source is stationary, so that for the sequences {s(t)} the time translation invariance holds: P(s(1); : : : ; s(N )) = P(s(t + 1); : : : ; s(t + N )). Now we introduce the N -block entropies HN = − P(CN ) ln P(CN ) ; (2.18) {CN }
and the di#erences hN = HN +1 − HN ;
(2.19)
whose meaning is the average information supplied by the (N + 1)th symbol, provided the N previous ones are known. One can also say that hN is the average uncertainty about the (N + 1)th symbol, provided the N previous ones are given. For a stationary source the limits in the following equations exist, are equal and de
(2.21)
The Shannon entropy is a measure of the “surprise” the source emitting the sequences can reserve to us, since it quanti<es the richness (or “complexity”) of the source. This can be precisely expressed by the
while
1 (N )
CN ∈
P(CN ) → 0
0 (N )
for N → ∞ :
(2.23)
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The meaning of this theorem is the following. An m-states process admits in principle mN possible sequences of length N , but the number of typical sequences Ne# (N ) (those ones in 1 (N )) e#ectively observable is Ne# (N ) ∼ exp(NhSh ) :
(2.24)
Note that Ne# mN if hSh ¡ ln m. Moreover the entropy per symbol, hSh , is a property of the source. Because of the ergodicity it can be obtained by analyzing just one single sequence in the ensemble of the typical ones, and it can also be viewed as a property of each one of the typical sequences. Therefore, as in the following, one may speak about the Shannon entropy of a sequence. The above theorem in the case of processes without memory is nothing but the law of large numbers. Let us observe that (2.24) is somehow the equivalent in information theory of the Boltzmann equation in statistical thermodynamics: S ˙ ln W , W being the number of possible microscopic con
(2.25)
In this way highly probable objects are mapped into short code words while the low probability ones are mapped to longer code words. So that the average code length is bounded by HN HN + 1 6 pr ‘(E(r)) 6 ; (2.26) ln 2 ln 2 r and in the limit N → ∞ one has pr ‘(E(r)) hSh
‘N lim = lim r = ; (2.27) N →∞ N N →∞ N ln 2 i.e., in a good coding, the mean length of a N -word is equal to N times the Shannon entropy (apart from a multiplicative factor, due to the fact that in the de
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An alternative coding method, based on variable length words, is due to Ziv and Lempel [138]. Remarkably it is very eJcient for data compression and gives the same asymptotic result of the Shannon–Fano code. 2.2.2. Again on the Kolmogorov–Sinai entropy After the introduction of the Shannon entropy we can give a more precise de
It is important to note that the probabilities P(W N (A)), computed by the frequencies of W N (A) along a trajectory, are essentially dependent on the stationary measure selected by the trajectory. This implies a dependence on this measure of all the quantities de
(2.31)
It is not simple at all to determine hKS according to this de
(2.32)
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We expect that h(Aj ) becomes independent of j when Aj is so
(2.33)
is, in some sense, more random than 1010101010101010101010 : : :
(2.34)
The notion of algorithmic complexity, independently introduced by Kolmogorov [126], Chaitin [53,56] and Solomonov [207], is a way to formalize the intuitive idea of randomness of a sequence. Consider a binary digit sequence (this does not constitute a limitation) of length N; (i1 ; i2 ; : : : ; iN ), generated by a certain computer code on some machine M. One de
(2.35)
where KU (N ) is the complexity with respect to the universal computer and CM depends only on the machine M. At this point we can consider the algorithmic complexity with respect to a universal computer—and we can drop the machine dependence in the symbol for the algorithmic complexity, K(N ). The reason is that we are interested in the limit of very long sequences, N → ∞, for which one de
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The bit length of the above program is O(ln N ) and therefore when taking the limit N → ∞ in (2.36), one obtains C = 0. Of course, K(N ) cannot exceed N , since the sequence can always be obtained with a trivial program (of bit length N ) “PRINT i1 ; i2 ; : : : ; iN ” :
(2.38)
Therefore, in the case of a very irregular sequence, e.g., (2.33) one expects K(N ) ˙ N , i.e. C= 0. In such a case one calls the sequence complex (i.e. of nonzero algorithmic complexity)
or random. Algorithmic complexity cannot be computed. Since the algorithm which computes K(N ) cannot have less than K(N ) binary digits and since in the case of random sequences K(N ) is not bounded in the limit N → ∞ then it cannot be computed in the most interesting cases. The un-computability of K(N ) may be understood in terms of GVodel’s incompleteness theorem [54 –56]. Beyond the problem of whether or not K(N ) is computable in a speci
N →∞
K(N )
HN
=
1 ; ln 2
(2.39)
where K(N ) = CN P(CN )KCN (N ), being KCN (N ) the algorithmic complexity of the N -words. Therefore the expected complexity K(N )=N is asymptotically equal to the Shannon entropy (apart the ln 2 factor). Eq. (2.39) stems from the results of the Shannon–McMillan theorem about the two classes of sequences (i.e. 1 (N ) and 0 (N )). Indeed in the limit of very large N , the probability to observe a sequence in 1 (N ) goes to 1, and the entropy of such a sequence as well as its algorithmic complexity equals the Shannon entropy. Apart from the numerical coincidence of the values of C and hSh =ln 2 there is a conceptual di#erence between the information theory and the algorithmic complexity theory. The Shannon entropy essentially refers to the information content in a statistical sense, i.e. it refers to an ensemble of sequences generated by a certain source. On the other hand, the algorithmic complexity de
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measure) one has hKS C(x) = ; (2.40) ln 2 where, as before, the factor ln 2 is a conversion factor between natural logarithms and bits. This result says that the KS-entropy quanti<es not only the richness, or surprise, of a dynamical system but also the diJculty of describing (almost) everyone of its typical sequences. 2.3. Algorithmic complexity and Lyapunov exponent Summing up, the theorem of Pesin together with those of Brudno and White show that a chaotic dynamical system may be seen as a source of messages that cannot be described in a concise way, i.e. they are complex. We expose here two examples that may help in understanding the previous conclusion and the relation between the Lyapunov exponent, the KS-entropy and the algorithmic complexity. Following Ford [79,80], let us consider the shift map x(t + 1) = 2 x(t)
mod 1 ;
(2.41)
which = ln 2. If one writes an initial condition in binary representation, i.e., x(0) = ∞ has −j , such that a = 0 or 1, it is clearly seen that the action of the map (2.41) on x(0) a 2 j j=1 j is just a shift of the binary coordinates: ∞ ∞ x(1) = aj+1 2−j · · · x(t) = aj+t 2−j : (2.42) j=1
j=1
With this observation it is possible to verify that K(N ) N for almost all the solutions of (2.41). Let us consider x(t) with accuracy 2−k and x(0) with accuracy 2−l , of course l=t+k. This means that, in order to obtain the k binary digits of the output solution of (2.41), we must use a program of length no less than l=t +k. Basically one has to specify a1 ; a2 ; : : : ; al . Therefore we are faced with the problem of the algorithmic complexity of the binary sequence (a1 ; a2 ; : : : ; a∞ ) which determines the initial condition x(0). Martin-LVof [156] proved a remarkable theorem stating that, with respect to the Lebesgue measure, almost all the binary sequences (a1 ; a2 ; : : : ; a∞ ), which represent real numbers in [0; 1], have maximum complexity, i.e. K(N ) N . In practice, no human being will ever be able to distinguish the typical sequence (a1 ; a2 ; : : : ; a∞ ) from the output of a fair coin toss. Finally, let us consider a 1d chaotic map x(t + 1) = f(x(t)) :
(2.43)
If one wants to transmit to a friend on Mars the sequence {x(t); t = 1; 2; : : : ; T } accepting only errors smaller than a tolerance , one can use the following strategy [174]: (1) Transmit the rule (2.43): for this task one has to use a number of bits independent of the length of the sequence T . (2) Specify the initial condition x(0) with a precision 0 using a
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(3) Let the system evolve till the
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the system. A simple argument previously suggested 1 : (3.1) Tp ∼ ln 0 However, in any realistic system, relation (3.1) is too naive to be of actual relevance. Indeed, it does not take into account some basic features of dynamical systems: • The Lyapunov exponent (2.9) is a global quantity: it measures the average rate of diver-
gence of nearby trajectories. In general, there exist
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(3.5) is piecewise linear and with a uniform invariant density, the explicit computation of L(q) is very easy. The moments of the response after a time t are simply given by q 1 1 q t q
R(t) = a + (1 − a) : (3.6) a 1−a From (3.3) and (3.6) we have L(q) = ln[a1−q + (1 − a)1−q ] ;
(3.7)
which recovers the non-intermittent limit L(q) = q ln 2 in the symmetric case a = 1=2. In the general case, assuming 0 6 a ¡ 1=2, we have that for q → +∞, L(q) is dominated by the less probable, most unstable contributions and L(q)=q −ln(a). In the opposite limit, q → −∞, we obtain L(q)=q −ln(1 − a). We now show how L(q) is related to the 8uctuations of R(t) at
(3.8)
In the limit t → ∞, the Oseledec theorem [169] assures that, for typical trajectories, +(t) = 1 = −a ln a − (1 − a) ln(1 − a). Therefore, for large t, the probability density of +(t) is peaked at the most probable value 1 . Let us introduce the probability density Pt (+) of observing a given + on a trajectory of length t. Large deviation theory suggests Pt (+) ∼ e−S(+)t ;
(3.9)
where S(+) is the Cramer function [216]. The Oseledec theorem implies that limt→∞ Pt (+) = (+ − 1 ), this gives a constraint on the Cramer function, i.e. S(+ = 1 ) = 0 and S(+) ¿ 0 for + = 1 . The Cramer function S(+) is related to the generalized Lyapunov exponent L(q) through a Legendre transform. Indeed, at large t, one has
q
R(t) = d+ Pt (+)eq+t ∼ eL(q)t ; (3.10) by a steepest descent estimation one obtains L(q) = max[q+ − S(+)] : +
In other words, each value of q selects a particular +∗ (q) given by dS(+) =q : d+ +∗
(3.11)
(3.12)
We have already discussed that, for negligible 8uctuations of the “e#ective” Lyapunov exponents, the LE completely characterizes the error growth and L(q) = 1 q. In presence of 8uctuations, the probability distribution for R(t) can be approximated by a log-normal distribution. This can be understood assuming weak correlations in the response function so that (3.2) factorizes in several independent contributions and the central limit theorem applies. We can thus write (ln R − 1 t)2 1 Pt (R) = ; (3.13) exp − 2/t R 2./t
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Fig. 1. Generalized Lyapunov exponent, L(q) for the map (3.5) with a = 0:3 (solid line) compared with the linear non-intermittent approximation, 1 q (dashed line), and with the log-normal one (Eq. (3.16), (dotted line)).
where 1 and / are given by 1
ln R(t) ; t 1 / = lim ( ln R(t)2 − ln R(t) 2 ) : t→∞ t
1 = lim
t→∞
(3.14)
The log-normal distribution for R corresponds to a Gaussian distribution for + S(+) =
(+ − 1 )2 ; 2/
(3.15)
and to a quadratic in q generalized Lyapunov exponent: L(q) = 1 q + 12 /q2 :
(3.16)
Let us remark that, in general, the log-normal distribution (3.13) is a good approximation for non-extreme events, i.e. small 8uctuation of + around 1 , so that the expression (3.16) is correct only for small q (see Fig. 1). This is because the moments of the log-normal distribution grow too fast with q [168]. Indeed from (3.12) we have that the selected +∗ (q) is given by +∗ (q) = 1 + /q and thus +∗ (q) is not
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Nevertheless, 1 and / are the two basic parameters for characterizing the intermittency of a system. To be more speci
(3.18)
Thus in the weak intermittency limit, /=1 ¡ 1, the most probable response function Rmax (t) follows the correct behavior (with the corrected exponent 1 − /). In the strong intermittent limit, /=1 ¿ 1, the most probable estimation breaks down because it predicts an asymptotic stable phase Rmax (t) → 0 instead of the chaotic exponential growth. As in the case of the
The generalized L(n) (q) represents the 8uctuations of the exponential divergence of a n-dimensional volume in phase space [172]. The properties of L(n) (q) are analogous to the properties of L(q), i.e. L(n) (q) is a concave function of q for any n and for a non-intermittent behavior they are linear in q. 3.2. Renyi entropies In Section 2.1.2 we de
where Aj is a partition of the phase space in cells of size j and W N (Aj ) indicates a sequence of length N in this partition. The generalized Renyi entropies [171,172], Kq , can be introduced by observing that (3.21) is nothing but the average of −ln P(W N ) with the probability P(W N ): 1 Kq = −lim lim lim P(W N (Aj ))q : (3.22) ln →0 j→0 N →∞ N(q − 1) N {W (Aj )}
As in (3.4) one has hKS =limq→1 Kq =K1 ; in addition from general results of probability theory, one can show that Kq is monotonically decreasing with q.
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It will not be surprising that the generalized Renyi entropies are related to the generalized Lyapunov exponents L(q). Introducing the number of non-negative Lyapunov exponents n∗ (i.e. n∗ ¿ 0, n∗+1 ¡ 0), the Pesin relation (2.16) can be written as ∗
hKS =
n i=1
∗ dL(n ) (q) i = : dq q=0
(3.23)
Moreover, one has [171] ∗
L(n ) (−q) Kq+1 = : −q
(3.24)
3.3. The e5ects of intermittency on predictability We have seen that intermittency can be described, at least at a qualitative level, in terms of 1 and /, which are the two parameters characterizing the log-normal approximation. We discuss now the relevance of the log-normal approximation for the predictability time Tp . The predictability time Tp is de
√
/w(t) ;
(3.26)
where w(t) is a Wiener process with w(0) = 0, w(t) = 0 and w(t)w(t ) = min(t; t ). In this case the computation of Tp is reduced to a
Tp is close to the most probable value of (3.27) corresponding to the naive estimation (3.1). On the contrary, in the strong intermittent limit, /=1 1, the pdf of Tp shows an asymmetric “triangular shape” and the most probable value is Tp =
1 ln(=0 )2 : 3/
(3.28)
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Fig. 2. Rescaled pdf, p(Tp )1, of the predictability time Tp for the Lorenz model (3.29): (a) with r = 28 (weak intermittency) the average predictability time is Tp =10:84 and its variance is 12 =3:12 while =0:90, /=0:06±0:01; (b) with r = 166:3 (strong intermittency) and Tp = 8:2385 and 12 = 19:75, while = 1:18 and / = 3:9 ± 1. The dashed line is the Gaussian distribution.
In order to see the e#ects of intermittency on the predictability time, let us consider as an example the Lorenz system [145]: dx = 1(y − x) ; dt dy = x(r − z) − y ; dt dz (3.29) = xy − bz ; dt with the standard values 1 = 10 and b = 83 . For r = 28, the Lorenz model is very weakly intermittent, /= 7 × 10−2 , and the pdf of the predictability time is very close to a Gaussian (Fig. 2). On the contrary, for r = 166:3 the Lorenz model becomes strongly intermittent [191], /= 3:3 and the pdf of the predictability time displays a long exponential tail responsible for the deviation from (3.1). This qualitative behavior is typical of intermittent systems. In Section 5.3 we will see a more complex example in the context of turbulence. 3.4. Growth of non-inBnitesimal perturbations In realistic situations, the initial condition of a system is known with a limited accuracy. In this case the Lyapunov exponent is of little relevance for the characterization of predictability and new indicators are needed. To clarify the problem, let us consider the following coupled map model: x(t + 1) = R x(t) + 4h(y(t)) ; y(t + 1) = G(y(t)) ;
(3.30)
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where x ∈ R2 , y ∈ R1 , R is a rotation matrix of arbitrary angle 5, h is a vector function and G is a chaotic map. For simplicity, we consider a linear coupling h(y) = (y; y) and the logistic map G(y) = 4y(1 − y). For 4 = 0 we have two independent systems: a regular and a chaotic one. Thus the Lyapunov exponent of the x subsystem is x (4 = 0) = 0, i.e., it is completely predictable. On the contrary, the y subsystem is chaotic with y = 1 = ln 2. If we now switch on the (small) coupling (4 ¿ 0) we are confronted with a single threedimensional chaotic system with a positive global Lyapunov exponent = y + O(4) : A direct application of (3.1) would give 1 ; Tp(x) ∼ Tp ∼ y
(3.31) (3.32)
but this result is clearly unacceptable: the predictability time for x seems to be independent of the value of the coupling 4. Let us underline that this is not due to an artifact of the chosen example. Indeed, one can use the same argument in many physical situations [32]. A well-known example is the gravitational three body problem with one body (asteroid) much smaller than the other two (planets). If one neglects the gravitational feedback of the asteroid on the two planets (restricted problem) one has a chaotic asteroid in the regular <eld of the planets. As soon as the feedback is taken into account (i.e. 4 ¿ 0 in the example) one has a non-separable three body system with a positive LE. Of course, intuition correctly suggests that it should be possible to forecast the motion of the planets for very long times if the asteroid has a very small mass (4 → 0). The apparent paradox arises from the use of (3.1), which is valid only for the tangent vectors, also in the non-in
(3.33)
where, with our choice, h = (y; y). At the beginning, both |x| and y grow exponentially. However, the available phase space for y is
(3.34)
so that Tp(x) ∼ 4−2 :
(3.35)
This example shows that, even in simple systems, the Lyapunov exponent can be of little relevance for the characterization of the predictability. In more complex systems, in which di#erent scales are present, one is typically interested in forecasting the large scale motion, while the LE is related to the small scale dynamics.
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Fig. 3. Growth of error |x(t)| for the coupled map (3.30). The rotation angle is 5 = 0:82099, the coupling strength 4 = 10−5 and the initial error only on the y variable is y = 0 = 10−10 . Dashed line |x(t)| ∼ e1 t where 1 = ln 2, solid line |x(t)| ∼ t 1=2 .
A familiar example is weather forecast: the LE of the atmosphere is indeed rather large due to the small scale convective motion, but (large scale) weather prediction is possible for about 10 days [146,160]. It is thus natural to seek for a generalization of the LE to
1 ln r : i (; r)
After having performed N error-doubling experiments, we can de
(; r) e
(3.36)
(3.37)
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Fig. 4. () as a function of for the coupled map (3.30) with 4 = 10−5 . The perturbation has been initialized as in Fig. 3. For → 0; () 1 (solid line). The dashed line shows the behavior () ∼ −2 .
where (; r) e is N
1
(; r) e = n (; r) : N
(3.38)
n=1
(see Appendix A and [12] for details). In the in
→0
(3.39)
In practice, this limit means that () displays a constant plateau at 1 for suJciently small (Fig. 3). For
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the computation of hKS involves the limit of arbitrary
(3.40)
Of course, X(m) (t) ∈ Rmd and it corresponds to the trajectory which lasts for a time T = m. In data analysis, the space where the state vectors of the system live is not known. Moreover, usually only a scalar variable u(t) can be measured. In such a case, one considers vectors (u(t); u(t + ); : : : ; u(t + m − )), that live in Rm and allow a reconstruction of the original phase space, known as delay embedding in the literature [209,199] (see also [1,2,114,170]), and it is a special case of (3.40). Introduce now a partition of the phase space Rd , using cells of edge j in each of the d directions. Since the region where a bounded motion evolves contains a
(3.41)
where i(j; t + j) labels the cell in Rd containing x(t + j). From the time evolution of X(m) (t) one obtains, under the hypothesis of ergodicity, the probabilities P(W m (j)) of the admissible words {W m (j)}. We can now introduce the (j; )-entropy per unit time, h(j; ) [201]: 1 (3.42) hm (j; ) = [Hm+1 (j; ) − Hm (j; )] ; 1 1 (3.43) h(j; ) = lim hm (j; ) = lim Hm (j; ) ; m→∞ m→∞ m where Hm is the block entropy of block length m: Hm (j; ) = − P(W m (j)) ln P(W m (j)) : (3.44) {W m (j)}
For the sake of simplicity, we ignored the dependence on details of the partition. To make h(j; ) partition-independent one has to consider a generic partition of the phase space {A} and to evaluate the Shannon entropy on this partition: hSh (A; ). The 4-entropy is thus de
inf
A:diam(A)64
hSh (A; ) :
(3.45)
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Note that the time dependence in (3.45) is trivial for deterministic systems, and that in the limit j → 0 one recovers the Kolmogorov–Sinai entropy hKS = lim h(j; ) : j→0
The above entropies Hm (j) have been introduced by using a partition and the usual Shannon entropy; however it is possible to arrive at the same notion, starting from other entropy-like quantities, that are numerically more convenient. For example, Cohen and Procaccia [61] proposed to estimate Hm (j) as follows. Given a signal composed of N successive records and the embedding variable X(m) , let us introduce the quantities: 1 nj(m) = 8(j − |X(m) (i) − X(m) (j)|) ; (3.46) N −m i =j
then the block entropy Hm (j) is given by 1 Hm(1) (j) = − ln nj(m) (j) : (N − m + 1) j
(3.47)
In practice, nj(m) (j) is an approximation of P(W m (j)). From a numerical point of view, correlation entropies [95,210] are sometimes more convenient, so that one studies (m) 1 Hm(2) (j) = −ln n (j) 6 Hm(1) (j) : (3.48) N −m+1 j j This corresponds to approximate the Shannon by the Renyi entropy of order q = 2 [114]. The (j; )-entropy h(j; ) is well de
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as inactive), excitation (i.e. the scale at which energy is injected in the system) and correlation (that we assume can be suitably de
(4.1)
j=−L=2
n with 40 = L=2 j=1 4j . L is the lattice size, i = −L=2; : : : ; L=2, x ∈ R is the state variable which depends on the site and time, and fa ∈ Rn → Rn is a non-linear map, which drives the local dynamics and depends on a control parameter a. Usually, periodic boundary conditions xi+L =xi are assumed and, for scalar variables (n=1), one studies coupled logistic maps, fa (x)=ax(1 − x) or tent maps, fa (x) = a| 12 − |x − 12 | |. The parameters {4i } rule the strength and the form of the coupling and they are chosen according to the physical system under investigation. For example, with 4j = 0 for j ¿ 2, i.e. nearest neighbor coupling, one can mimic PDEs describing reaction di#usion processes (indeed formally the equation assumes the structure of a discrete Laplacian). However, it could be misleading to consider CMLs simply as discrete approximation of PDEs. Indeed, since the local map fa is usually chaotic, chaos in CML, di#erently from PDE, is the result of many interacting chaotic sub-systems. Hence, the correspondence between the instability mechanisms in the two type of models is not straightforward [68].
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Other kinds of coupling can be considered to mimic di#erent physical situations, e.g., asymmetric CML (see Section 4.5) for studying convective instabilities [106,39,217], or mean <eld (globally coupled maps) version of (4.1) (Section 4.3) for studying neural network or population dynamics [118]. Further generalizations are quoted in Ref. [112]. Lyapunov exponents, attractor dimensions and entropies can be de
L→∞
(4.2)
where x = i=L is a continuous index in [0; 1], and <(x) is a non-increasing function. The function <(x) can be viewed as a density of Lyapunov exponents. If such limit does not exist, the possibility to build a statistical description of spatio-temporal chaos would be hopeless, i.e., the phenomenology of these systems would depend on L. Once the existence of a Lyapunov density is proved, one can generalize some results of low-dimensional systems [97,44], namely the Kaplan–Yorke conjecture [117] and the Pesin relation (2.16). For instance, one can generalize (2.16) to
1 hKS HKS = lim d x <(x)5(<(x)) (4.3) = L→∞ L 0 5(x) being the step function. In the same way one can suppose the existence of a dimension density DF , that is to say a density of active degrees of freedom, i.e. DF = limL→∞ DF =L which by the Kaplan–Yorke [117] conjecture is given by [97]
DF d x <(x) = 0 : (4.4) 0
The existence of a good thermodynamic limit is supported by numerical simulations [109,144] and some exact results [205]. Recently, Eckmann and Collet [62] have proved the existence of a density of degrees of freedom in the complex Ginzburg–Landau equation. See also Refs. [97,44] and references therein for a discussion on such a problem. 4.2. Overview on the predictability problem in extended systems In low-dimensional systems, no matter how the initial disturbance is chosen, after a—usually short—relaxation time, TR , the eigendirection with the largest growth rate dominates for almost all the initial conditions (this, e.g., helps in the numerical estimates of the Lyapunov exponents [20]). On the contrary, in high-dimensional systems this may not be true [92,183,173,186]. Indeed, in systems with many degrees of freedom there is room for several choices of the initial perturbation according to the speci
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certain degrees of freedom or homogeneous in all the degrees of freedom), and it is not obvious that for all of them the time TR needed to align along the maximally expanding direction is the same. In general, the situation can be very complicated. For instance, it is known that, also considering initially homogeneous disturbances, the Lyapunov vectors can localize (permanently or slowly wandering) on certain degrees of freedom [109,75,186]. Of course, this will severely a#ect the prediction of the future evolution of the system. Indeed, regions of large predictability time could coexist with regions of relatively short predictability time. In Ref. [109,112,186] one
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Tp ∼ O(1=). However, since usually such phenomena arise in <elds, whose evolution is ruled by PDE, the non-equivalence of the norms makes a general approach to the problem unfeasible. Therefore, one has to resort to ad hoc treatments, based on physical intuition to identify the most suitable norm to be used for the particular needs (see Section 4.8). 4.3. ButterCy e5ect in coupled map lattices In spatially extended systems it is important to understand the way an uncertainty initially localized in some region will spread. Here we study in particular the time needed for a perturbation, initially seeded in the central site of a lattice of coupled maps, to reach a preassigned value at the border of the lattice [173] (see also [109,212,213] and Section 4.6). In other terms we wonder about the “butter8y e#ect” starting from the center of the lattice and arriving up to the boundary. We shall discuss the properties of such time by varying the coupling range from local to non local in the 1-dimensional CML (4.1) with periodic boundary conditions. We consider two cases: local coupling, i.e. 4j = 0 if j ¿ 2, and non-local coupling, e.g. C2 (4.5) 41 = C1 and 4j = = for j ¿ 2 ; j where = measures the strength of non-locality. The initial perturbation is on the central site, i.e. |xi (0)| = 0 i; 0 :
(4.6)
We look at the predictability time Tp needed for the perturbation to reach a certain threshold max on the boundary of the lattice, i.e. the maximum time, t, such that |xL=2 (t)| 6 max . For nearest neighbor coupling, one has obviously that xL=2 (t) = 0 for t ¡ L=2. Indeed, by a numerical integration of (4.1) for the short range coupling one observes that xL=2 (t) = 0 for times t ¡ t ∗ ˙ L; while for t ¿ t ∗ the perturbation, due to the (local) chaotic dynamics, grows as xL=2 (t) ∼ 0 exp[(t − t ∗ )]. Thus for local interactions, the predictability is mainly determined by the waiting time t ∗ , necessary to have |xL=2 | ¿ 0 , which is roughly proportional to the system size L. This is con
Tp = t1 + GL ;
(4.7)
−1
where the time t1 ∼ is due to the exponential error growth after the waiting time and can be neglected in large enough lattices. This agrees with the existence of a
Tp ∼ t1 ∼ −1 :
(4.8)
As shown in Fig. 5, weakly non-local couplings, and mean <eld interactions (4j = C2 =N ) have the same qualitative behavior. Very accurate numerical computations have con
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Fig. 5. Average predictability time Tp versus L for a CML of logistic maps fa (x) = ax(1 − x) with a = 4: local coupling j0 = 0:3 (squares); non-local coupling (4.5) with C1 = 0:3, C2 = 0:01 = = 2 (crosses) or = = 3 (diamonds); mean <eld coupling ji = C2 =L with C2 = 0:3 (crosses squares). The initial perturbation is applied at the center of the lattice (site i = 0) and has an amplitude 10−14 ; the maximum admitted error is max = 0:1.
This example demonstrates that the predictability time is given by two contributions: the waiting time t ∗ and the characteristic time t1 ∼ −1 associated with chaos. For non-local interactions, the waiting time practically does not depend on the system size L, while for local interactions it is proportional to L. Let us underline that in these results the non-linear terms in the evolution of x(t) are rather important. One numerically observes that the waiting time t ∗ is not just the relaxation time TR of x on the tangent eigenvector. Actually, TR is much larger than t ∗ . 4.4. Comoving and speciBc Lyapunov exponents A general feature of systems evolving in space and time is that a generic perturbation not only grows in time but also propagates in space. Aiming at a quantitative description of such phenomena, Deissler and Kaneko [71] introduced a generalization of the LE to a non-stationary frame of reference: the comoving Lyapunov exponent. For the sake of simplicity, we consider again the case of a 1-dimensional CML. Let us consider an in
where (v) is the largest comoving Lyapunov exponent, i.e. |x[vt] (t)| 1 lim : (v) = lim lim ln t→∞ L→∞ |x0 (0)|→0 t |x0 (0)|
(4.9) (4.10)
In Eq. (4.10) the order of the limits is important to avoid boundary e#ects. For v=0 one recovers the usual LE. Moreover, one has that (v) = (−v) (and the maximum value is obtained at v = 0 [189]) when a privileged direction does not exist, otherwise (v) can be asymmetric and the
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maximum can be attained at value v = 0 (see Section 4.6). By writing the response function (2.8) in the moving frame one can also introduce the generalized comoving Lyapunov exponents Lq (v) for studying
i = [vt] :
(4.12)
Note that for / = 0, <(/) reduces to the standard LE. Therefore, the comoving Lyapunov exponent is given by d<(/) (v) = <(/) − / : (4.13) d/ The last equation de
(4.14)
where t and n (=1; : : : ; L) are the discrete time and space respectively; the map fa (x) is usually chosen to be the logistic map. One can consider di#erent boundary conditions, x0 (t). For instance, x0 (t) = x∗ with x∗ being an unstable <xed point of the map fa (x), or more generic time-dependent boundary conditions where x0 (t) is equal to a known function of time y(t), which can be periodic, quasi-periodic or chaotic. Here, following Pikovsky [180 √ –182], we consider a quasi-periodic boundary condition x0 (t)=0:5+0:4 sin(!t), with !=.( 5 − 1). However,
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Fig. 6. Sketch of the behavior of (v) for (a) an absolutely and convectively stable 8ow, (b) absolutely stable but convectively unstable 8ow, and (c) absolutely unstable 8ow.
the results we are going to discuss do not depend too much on the details of the boundary conditions, i.e. on using x0 (t) quasi-periodic or chaotic. A central concept in the study of 8ow systems is the one of convective instability, i.e. when a perturbation grows exponentially along the 8ow but vanishes locally. We may give a description of the phenomenology of 8ow systems in terms of the largest LE and of the comoving LE. The absolute stability is identi<ed by the condition (v) ¡ 0 for all v ¿ 0; the convective instability corresponds to 1 = (v = 0) ¡ 0 and (v) ¿ 0 for some velocities v ¿ 0 and
(v) x n (t) ∼ x0 (t − )e dv = 0 e[(v)=v]n dv : (4.15)
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Fig. 7. Evolution of a state of the system (4.14) where fa (x) is the logistic maps, the boundary condition is quasi-periodic, a = 3:85 and c = 0:7: in this case 1 ¡ 0 but the system is convectively unstable.
Since we are interested in the asymptotic spatial behavior, i.e. large n, we can write x n (t) ∼ 0 e@n ;
(4.16)
The quantity @ can be considered as a sort of spatial-complexity index, an operative de
x n (t) ∼ 0 e[+˜t (v)=v]n dv ; (4.19) and therefore
1 |x n | 1 |xtypical | +˜t (v) n ln = lim ln = max : @ = lim n→∞ n n→∞ n v 0 0 v
(4.20)
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Fig. 8. @(+) and @∗ () vs. a at a <xed value of c (=0:7), for the system (4.14) of logistic maps with quasi-periodic boundary conditions (the system is convectively unstable for all the considered values of the parameters).
Therefore, because of the 8uctuations, it is not possible to write @ in terms of (v), although one can obtain a lower bound [217]: (v)
+˜t (v) @ ¿ max ≡ @∗ : = max (4.21) v v v v In Fig. 8 we show @ and @∗ vs. a for a <xed value of c. There is a large range of values of the parameter a for which @ is rather far from @∗ . This di#erence is only due to intermittency, as investigations of the map fa (x) = ax mod 1 or the computation of the generalized spatial Lyapunov exponents Ls (q) [217] con
(4.22)
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Fig. 9. (a) Space–time evolution of |xi (t)| for an initially localized perturbation (4.6) with 0 = 10−8 . We used a CML of tent maps, fa (x) = a(1=2 − |x − 1=2|), with a = 2; j = 2=3 and L = 1001. (b) (v) for v ¿ 0 for the CML of (a). The straight line indicates the zero and the intersection between the curve (v) and 0 indicates the perturbation velocity VF ≈ 0:78.
The interesting point in Eq. (4.22) is that it gives not only a de
d − d/ /
=−
(v) : /2
(4.23)
Moreover, since (VF ) = 0 (4.22) one has that dV=d/ = 0 at /0 such that V (/0 ) = VF , i.e. /0 selects the minimal velocity. Indeed <(/) is convex (being a Legendre transform), so that the
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Fig. 10. max [(; v)] (dashed line with points) versus v for the shift map f(x) = rx mod 1 with r = 1:1 and j0 = 13 , compared with (v) (continuous line). The two vertical lines indicates the velocity obtained by (4.22) which is about 0:250 and the directly measured one VF ≈ 0:342. Note that max [(; v)] approaches zero exactly at VF .
minimum is unique and
<(/0 ) d<(/) VF = = : /0 d/ /=/0
(4.24)
Thus for an in
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Eq. (4.22), which is able to take into account also possible non-linear e#ects, is max[(; VF )] = 0 :
The numerical evidences also suggest that the condition which should be accomplished in order to have deviation from the linear prediction given by (4.22) and (4.24) is that (; v = 0) ¿ (0; 0) = , con
4 x n (t + 1) = (1 − 4)fa (x n (t)) + fa (xi (t)) ; N
(4.25)
i=1
where N is the total number of elements. The evolution of a macroscopic variable, e.g., the center of mass m(t) =
N
1 xi (t) ; N
(4.26)
i=1
upon varying 4 and a in Eq. (4.25), displays di#erent behaviors [50]: (a) Standard chaos: m(t) obeys a Gaussian statistics with a standard deviation 1N =
m(t)2 − m(t) 2 ∼ N −1=2 .
(b) Macroscopic periodicity: m(t) is a superposition of a periodic function and small 8uctuations O(N −1=2 ). (c) Macroscopic chaos: m(t) displays an irregular motion as it can be seen by looking at the plot of m(t) vs. m(t − 1) that appears as a structured function (with thickness ∼ N −1=2 ), and suggests a chaotic motion for m(t). Phenomena (a) and (b) also appear in CML with local coupling in high enough dimensional lattices [58], for the interesting case (c), as far as we know, there is not a direct evidence in
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a hydro-dynamical (coarse grained) level, i.e.: regular motion (hydro 6 0) or chaotic motion (0 ¡ hydro 1 ). In principle, if one knows the hydrodynamic equations, it is possible to characterize the macroscopic behavior by means of standard dynamical system techniques. However, in generic CML there are no general systematic methods to build up the macroscopic equations, apart from particular cases [113,184]. Therefore, here we discuss the macroscopic behavior of the system relying upon the full microscopic level of description. The microscopic Lyapunov exponent cannot give a characterization of the macroscopic motion. To this purpose, recently di#erent approaches have been proposed based on the evaluation of the self-consistent Perron–Frobenius (PF) operator [113,178,184] and on the FSLE [50,203]. Despite the conceptual interest of the former (in some sense the self-consistent PF-operator plays a role similar to that of the Boltzmann equation for gases [50]), here we shall only discuss the latter which seems to us more appropriate to address the predictability problem. We recall that for chaotic systems, in the limit of in
Fig. 11. () versus for the system (4.25) with a = 1:7; j = 0:3 for N = 104 (×); N = 105 ( ); N = 106 ( ) and N =107 (). The
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one has another plateau, the “macroscopic” Lyapunov exponent, () = M . Moreover, 1 √and 2 decrease at increasing N : indeed, by looking at Fig. 11b one can see that 1 ; 2 ∼ 1= N . It is important to observe that the macroscopic plateau, being almost non-existent for N = 104 , √ becomes more and more resolved and extended on large values of N at increasing N up to N = 107 . Therefore we can argue that the macroscopic motion is well de
(4.27)
where is the stream function such that v = (9y ; −9x ) and = −!. As customary in direct numerical simulations, the dissipation is modi<ed by employing high order viscosity p ¿ 1 in order to achieve larger Reynolds numbers. The numerical results discussed below are obtained
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by integrating (4.27) by means of a standard pseudo-spectral code on a periodic computational domain with resolution N × N . The classical theory of predictability in turbulence [136,137] studies the evolution of a difference (or error) <eld, de
∞ 1 Z (t) = d 2 x |!(x; t)|2 = d k Z (k; t) ; (4.29) 2 0
E (t) =
∞
0
d k k −2 Z (k; t) =
∞
0
d k E (k; t) ;
(4.30)
where we have also introduced the enstrophy (Z ) and energy (E ) error spectra. It is also natural to introduce the relative errors, de
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Fig. 12. Relative energy (r) and enstrophy (z) error growth for a 5122 simulation. Tp indicate the predictability time de
Fig. 13. Gray scale map of the vorticity <elds (obtained by a 2562 simulation) at time Tp = 177. White corresponds to positive vorticity regions, black to negative ones. (a) Reference <eld !(x), (b) the perturbed one ! (x), (c) the error
error curves bend and a predictability time estimation with energy norm gives Tp 395. From Fig. 12 we learn at least two lessons. First (but not surprisingly) about half of the predictability time is governed by non-exponential error growth behavior. This is another demonstration of the little relevance of LE for characterizing predictability in realistic complex systems. The second observation is that the di#erent norms r(t) and z(t) give qualitatively similar results. Because the error is initially con
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about 65% in enstrophy, they look still remarkably similar for what concerns the distribution of vortices. Most of the large coherent structures are almost in the same positions. In Fig. 13 we also plot the di#erence <eld !(x; Tp ). The typical bipolar con
Fig. 14. Mean vortex separation d(t) at resolution 5122 . At the classical predictability time Tp , the mean vortex separation is about one-tenth of the saturation level.
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time, the mean vortex separation is d(Tp ) 0:5, well below the saturation value (dmax ∼ L=2=. in the periodic computational box). This result is a quantitative con
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Fig. 15. Typical turbulent energy spectrum, kL−1 is the energy containing integral scale and kD−1 the dissipative Kolmogorov scale.
related to the smallest active eddies. The appearance of a power law in between these two extremes unveils that no other characteristic scale is involved. A simple and elegant explanation of these experimental
(5.1)
The original Kolmogorov theory (K41) assumes self-similarity of the turbulent 8ow. As a consequence, the scaling behavior of higher-order structure functions Sp (‘)= |v(x+‘)−v(x)|p ∼ ‘Fp is described by a single scaling exponent. The value of the exponent is determined by the so-called “4=5 law”, an exact relation derived by Kolmogorov from the Navier–Stokes equations [123,84], which, under the assumption of stationarity, homogeneity and isotropy states
v 3 (‘) = − 45 jW‘ ;
(5.2)
where v (‘) is the longitudinal velocity di#erence between two points at distance ‘, and jW is the average rate of energy transfer. The structure function exponent Fp is thus predicted by Kolmogorov similarity theory to be Fp = p=3. Several experimental investigations [7,84] have shown that the Kolmogorov scaling is not exact and Fp is a non-linear function (with F3 =1 as a consequence of the “4=5 law”). This means a breakdown of the self-similarity in the turbulent cascade. Larger and larger excursions from mean values are observed as one samples smaller and smaller scales. This phenomenon goes under the name of intermittency [84]. A complete theoretical understanding of intermittency in
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Navier–Stokes turbulence is still lacking. Nevertheless, there are approaches, as the multifractal model [177], which are able to characterize at a phenomenological level the intermittency. In brief, the basic idea of the multifractal model [177,172,84] consists in assuming a local scale-invariance for the velocity 8uctuations, i.e. one has v‘ ∼ ‘h , with a continuous spectrum of (HVolder) exponents h, each belonging to a given fractal set. In other words, in the inertial range one has v‘ (x) ∼ ‘h ; (5.3) if x ∈ Sh , and Sh is a fractal set with dimension D(h) and h ∈ (hmin ; hmax ). The probability to observe a given scaling exponent h at the scale ‘ is thus P‘ (h) ∼ ‘3−D(h) . In this language the Kolmogorov similarity theory [123,84] corresponds to the case of only one singularity exponent h = 13 with D(h = 13 ) = 3, see also Appendix B. 5.2. Reduced model of turbulence In numerical simulations of the Navier–Stokes equations in the regime of fully developed turbulence, one has to discretize the original PDE to obtain a set of approximate ODE which must be integrated numerically. This is the direct numerical simulation approach which, in its simplest form, is implemented on a regular 3D grid of N 3 points. Since the dissipative scale (Kolmogorov scale) is related to the Reynolds number as ‘D ∼ LRe−3=4 , an estimate of the number N of active spatial degrees of freedom leads to N ∼ (L=‘D )3 ∼ Re9=4 :
(5.4) An obvious consequence of the fast growth of N with the Reynolds number is the unfeasibility of a complete turbulent simulations at high Re. The maximum limit of present computers is about N =103 which corresponds to Re 104 . An alternative approach has been introduced with the so-called shell models by the works of Obukhov, Gledzer and Desnyansky and Novikov (see [38] for a detailed discussion). The basic idea, originally motivated in the context of closure theory, is to implement a dynamical cascade with a set of variables un (n = 1; : : : ; N ) each representing the typical magnitude of the velocity 8uctuation in a shell of wave-numbers kn ¡ |k| ¡ kn+1 . The representative wave-numbers are spaced geometrically, kn = k0 2n , in this way, assuming locality in the cascade, interactions are con
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The number of shells N necessary to mimic the cascade mechanism of fully developed turbulence is rather small, due to the geometrical progression in kn one has N ∼ log2 Re. We have thus a chaotic dynamical system with a reasonably small number of degrees of freedom where standard methods of deterministic chaos can be used in order to relate the “turbulent” statistical description in terms of structure functions and intermittency, and dynamical properties, such as the spectrum of Lyapunov exponents. The absence of any stochastic term in (5.5) makes the shell model a natural model for investigating the predictability problem in turbulence. 5.3. E5ects of intermittency on predictability of inBnitesimal perturbations The sensitive dependence on initial conditions makes the long term forecasting in turbulent 8ow practically impossible. For instance, Ruelle [196] remarked that thermal 8uctuations in the atmosphere produces observable changes on a scale of centimeters after only few minutes. As a consequence after one or two weeks, the large-scale atmospheric circulation would be completely unpredictable, even if the exact evolution equations were known. This is the so-called butterCy e5ect, in the words of Lorenz: A butterCy moving its wings in Brazil might cause the formation of a tornado over Texas. To support this argument, one can observe that the largest LE of fully developed turbulence is roughly proportional to the inverse of the smallest characteristic time of the system, the turn-over time D of eddies of the size of the Kolmogorov length ‘D . From ‘D ∼ LRe−3=4 one obtains D ∼ ‘D =vD ∼ L Re−1=2 ;
(5.6)
where L ≈ L=U is the eddy turn-over time of the energy containing scales. As a consequence, as
(5.7)
(see Appendix B for details). To obtain the largest Lyapunov exponent now we have to integrate D (h)−1 , at the scale ‘ = ‘D (h), over the h-distribution P‘ (h) ∼ ‘3−D(h) :
h−D(h)+2
1 ‘D −1 dh : (5.8) ∼ (h) P‘ (h) dh ∼ L L Since the viscous cut-o# vanishes in the limit Re → ∞, the integral can be estimated by the saddle-point method, i.e. 1 = D(h) − 2 − h ∼ Re with = = max : (5.9) h L 1+h
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Fig. 16. Lyapunov exponent ( ) and variance / (×) as a function of the Reynolds number Re for the shell model (5.5) with N = 27 shells. The dashed line is the multifractal prediction ∼ Re= with = = 0:459, with function D(h) obtained by the random beta model
The value of = depends on the shape of D(h). By using the function D(h) obtained by
(5.10)
where in the last expression we have introduced the integral correlation time tc = C(t) dt of the e#ective Lyapunov exponent [66,38], where C(t) is the normalized correlation function of the 8uctuation of +(t) (i.e. +(t) − ). The quantity (+ − )2 can be computed by repeating the argument for : 1
+2 ∼ −2 ∼ 2 Rey : (5.11) L
An explicit calculation [66] gives y = 1 independently of intermittency. Assuming that the correlation time tc vanishes as a power of Re tc ∼ L Re−z
(5.12)
one ends with the prediction 1 (5.13) / ∼ Rew with w = 1 − z : L Numerical simulations on the shell model (5.5) give w 0:8 (see Fig. 16). Because w ¿ = we obtain that /= diverges with Re. From Fig. 16 we see that the strong intermittency regime begins, for the shell model, at Re ∼ 106 . Let us stress that in the absence of intermittency one would expect that tc ∼ −1 and thus z = 12 and /= constant. The fact that z ∼ 0:2 indicates that
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Fig. 17. Rescaled probability distribution functions of the predictability time Tp for the shell model (5.5) for (a) Re = 106 and (b) Re = 2 × 109 . The respective average values are Tp = 84:0 and 6:32 and the standard deviations are 1(Tp ) = 22:2 and 3:16. The line is the Gaussian.
the presence of quiescent periods in the turbulent activity is much more relevant for the decay rate of time correlations than for the Lyapunov exponent. We have seen in Section 3.3 that the 8uctuations of the e#ective LE a#ect the distribution of predictability time, and thus we expect a similar e#ect in fully developed turbulence. In the shell model one can estimate the predictability time by computing the time Tp at which the di#erence um (t) (where m corresponds to the integral scale) among two realizations of the model becomes larger that the tolerance . The initial di#erence 0 is restricted to the shell un∗ on the Kolmogorov scale and mn∗ . The predictability time distribution function is computed at two di#erent Reynolds number. At Re = 106 we are at the border of the weak intermittent range: the observed PDF (Fig. 17) is indeed close to a Gaussian with mean value 1
Tp ln : (5.14) 0 On the contrary, at Re = 2 × 109 , the PDF exhibits the asymmetric triangular shape and the mean value is ruled by / according to (3.28). 5.4. Growth of non-inBnitesimal perturbations The classical theory of predictability in turbulence has been developed by Lorenz [146] (see also [143]) using physical arguments, and by Leith and Kraichnan [137] on the basis of closure approximations. The fundamental ingredients of the Lorenz approach stem from dimensional arguments on the time evolution of a perturbation in an energy cascade picture. In this framework, it is rather natural to assume that the time ‘ for a perturbation at scale ‘=2 to induce a complete uncertainty on the velocity <eld on the scale ‘, is proportional to the typical eddy turn-over time at scale ‘: ‘ ∼ ‘=v‘ where v‘ is the typical velocity di#erence at scale
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‘. Kolmogorov scaling (5.1) gives ‘ ∼ ‘2=3 :
(5.15)
Because of the geometric progression (5.15), the predictability time to propagate an uncertainty O(vD ) from the Kolmogorov scale ‘D up to the scale of the energy containing eddies L, is dominated by the longest time L Tp ∼ ‘d + 2‘d + · · · + L ∼ L ∼ : (5.16) vL Closure approximations, where one still uses dimensional arguments, con
(5.17)
In the dissipative range ¡ vD , the error can be considered in
−1 () ∼ L dh(=vL )[3−D(h)]=h (=vL )1−1=h : (5.18) From the basic inequality of the multifractal model D(h) 6 3h + 2 (see Appendix B), we have 2 + h − D(h) ¿ −2 for all h : (5.19) h As a result of the constancy of the energy 8ux in the inertial range, 4W = v3 (‘)=‘, the equality holds for h = h∗ (3), and gives 3h∗ (3) + 3 − D(h∗ (3)) = 1. Therefore a saddle-point estimation of (5.18) gives again (5.17). The dimensional scaling of the FSLE in fully developed turbulence () ∼ −2 is thus not a#ected by intermittency corrections. This is a direct consequence of the exact result (5.2). These
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Fig. 18. The inverse of the error doubling times versus u (diamond) compared with shell turn-over times (plus). Number N of simulated shells is 27, and Reynolds number Re = C−1 = 109 , k0 = 0:05. The initial perturbation is randomly uniform over all shells in the inertial range, with amplitude of order 10−6 . The
Let us observe that, even at this high Reynolds number, the scaling range for the doubling time is rather small. It is interesting to look at the doubling time as a function of the Reynolds number. For small thresholds the inverse of the doubling time scales as the Lyapunov exponent, i.e. roughly as Re−1=2 . We also observe that the bend away from the in
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Fig. 19. ln[1=(u; r)=Re1=2 ] versus ln[u=Re−1=4 ] at di#erent Reynolds numbers Re=C−1 . () N =24 and Re=108 ; (+) N = 27 and Re = 109 ; ( ) N = 32 and Re = 1010 ; (×) N = 35 and Re = 1011 . The straight line has slope −2.
Fig. 20. ln1=(u; r)=ln(Re=Ro ) versus ln(u=Vo )=ln(Re=Ro ); multiscaling data collapse at di#erent Reynolds numbers Re = C−1 . The
suggests that the dimensional scaling (5.17) is observable even in direct numerical simulations at moderate Reynolds number. Let us consider two realizations of the vorticity <eld in (4.27) starting from very close initial conditions. The error is de
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Fig. 21. Finite size Lyapunov exponent () as a function of velocity uncertainty in a direct numerical simulations with 10242 grid points of 2D turbulence in the inverse cascade regime. The asymptotic constant value for → 0 is the largest Lyapunov exponent of the turbulent 8ow. The dashed line has slope −2. In the inset we show the compensated plot ()2 = jW.
of intermittency, also the crossover from the in
(5.20)
where G is an adimensional constant. It is easy to realize √ that (5.20) is equivalent to (5.17), () having the dimension of an inverse time and = E . The result obtained in numerical simulations is shown in Fig. 22, which has to be compared with Fig. 21. The scaling law (5.20) in Fig. 22 is barely visible, making the determination of G diJcult. On the contrary, inverting (5.17) to (5.20) one can measure G directly from Fig. 21. The result obtained is in close agreement with closure computations [37]. 5.5. j-entropy for turbulent Cows A complementary way to look at the predictability of turbulent 8ows is in terms of its entropy (see Sections 2:1:2 and 3:5). Unfortunately, a direct measurement of the Kolmogorov–Sinai entropy is practically infeasible. Indeed for Re → ∞ due to the huge number of active degrees of freedom, the KS-entropy diverges, so that one needs velocity measurements with an extremely high resolution and lasting for extremely long times, far beyond the actual experimental possibilities. Nevertheless, limiting the analysis to not very high resolution, one can hope to extract some interesting piece of
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Fig. 22. Average energy error E (t) growth. Dashed line represents linear closure prediction, dotted line is the saturation value E. The initial exponential growth is emphasized by the lin-log plot in the inset.
information by investigating the behavior of the j-entropy, h(j). As far as the j-entropy of turbulence is concerned, two questions can be raised. (i) Since a direct measurement of the full 3-dimensional velocity <eld is infeasible, one has usually access just to a time signal measured in one spatial point: which kind of information can we extract from the j-entropy per unit time of such a signal? (ii) Taking into account (i), can we say something about the j-entropy of the full 3-dimensional velocity <eld? In (ii) we are referring to the j-entropy, hST (j), per unit time and volume (the symbol ST means space–time). In other words, we are assuming that the total entropy of a turbulent 8ow observed for a (very long) time T on a (very large) volume V of the 3-dimensional space has the form H (V; T; j) ≈ VThST (j). See Ref. [89] for an introduction of this concept. Both in (i) and (ii), as we will see, a crucial role is played by the sweeping of the large scales of the 8ow on the small ones, i.e. the Taylor hypothesis (see Section 5.1). 5.5.1. j-entropy for a time signal of turbulence In order to estimate the j-entropy of a given signal one has to compute the Shannon entropy of the symbolic sequence obtained by making an (j; ) grid in phase-space (Section 3.5). Unfortunately, this method is rather ineJcient for signals in which many scales are excited [3,4,51], e.g., as in turbulence. Therefore, here we resort to a recently proposed method [3] based on the exit-time analysis. In a few words, the idea consists in looking at a sequence of data not at <xed sampling times but at <xed 8uctuation (see Appendix C), i.e. when the 8uctuation of the signal exceeds a given threshold, j. In practice, we code the signal v(t) of total duration T ina symbolic i−1 sequence M (j) = {ti (j); ki (j)}M i=1 , where ti (j) is the
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M (j) is a faithful coding of the the total number of exit events, i.e. M i=1 ti (j) = T . Note that signal within the required accuracy j. Now the evaluation of the entropy goes as usual through the evaluation of the Shannon entropy, h (j), of the sequence M (j). Finally, the j-entropy per unit time is given by [3] h (j; r ) h(j) ≈ ; (5.21)
t(j) where a coarse-graining of the possible values assumed by t(j) with a resolution time r has been considered, and t(j) is the average exit time, i.e. t(j) = (1=M ) i=1; M ti (j). Formula (5.22) is exact in the limit r → 0 (in Appendix C one
t(j) ∼ j1=h
with P(h) ∼ j(3−D(h))=h :
(5.22)
The exit-time moments [30], also called inverse structure functions [107], can be estimated in the multifractal framework as follows
q Tt (j)U ∼ dh j(q+3−D(h))=h ∼ jI(q) ; (5.23) where I(q) may be obtained with a saddle-point estimate in the limit of small j: q + 3 − D(h) : (5.24) I(q) = min h h The average [ : : : ] , obtained by counting the number of exit-time events M , and the average T[ : : : ]U, computed with the uniform time sampling are connected by the relation M ti
t q+1 (j) Tt q (j)U = lim tiq M = ; (5.25) M →∞
t(j) t j j=1 i=1 M where the term ti = j=1 tj takes into account the non-uniformity of the exit-time statistics. Therefore the quantity we are looking for, i.e. the mean exit-time, is given by t(j) = Tt −1 (j)U−1 ∼ (j)−I(−1) . By noting that −1 + 3 − D(h) 2 − D(h) ¿ −3 for all h ; (5.26) = h h which is nothing but Eq. (5.19), i.e. the 45 law of turbulence, we
(5.27)
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◦
Fig. 23. Numerically computed lower bound ( ) and upper bound ( ), with = 0:1t(j) for the (j; )-entropy in the case of a multiaJne signal with F(3) = 1. The signal has been obtained with the method of Ref. [29] (see also Appendix D) using a D(h) which
In Fig. 23 we report the evaluation of the upper and lower bounds (see Appendix C) of h(j) for a synthetic signal, v(t), constructed in such a way as to reproduce the statistical properties of turbulence [29]. Let us now compare the above results with a previous study of the j-entropy in turbulence [224], where it was argued that h(j) ∼ j−2 ;
(5.28)
a behavior that di#ers from the prediction (5.27). The behavior (5.28) has been obtained by assuming that h(j), at scale j, is proportional to the inverse of the typical eddy turnover time at that scale: since the typical eddy turnover time for velocity 8uctuations of order v ∼ j is (j) ∼ j2 , Eq. (5.28) follows. Indeed this is the argument used to derive (5.17) for the FSLE. The di#erence between (5.28) and (5.27) can be understood by considering that even if () and h(j) are two complementary concepts (the fact that for both the estimate of the scaling behavior reduces to the “4=5 law” is not a coincidence), in the latter case one has to consider the sweeping induced by the large scales. On the contrary, since the former is related to the distance of two realizations which di#er in the small scales (¡ ) but not on the large scales (¿ ), the sweeping of the large scales is not e#ective. 5.5.2. j-entropy of turbulence and the Taylor hypothesis Now we study the j-entropy per unit time and volume for the velocity <eld of turbulent 8ows in 3 + 1 dimensions, hST (j). We will show that, by assuming the usually accepted Taylor hypothesis, one has a spatial correlation which can be quantitatively characterized by an “entropy” dimension D = 83 . As already remarked, hST (j) cannot be directly measured so we will discuss its estimation in a theoretical framework by introducing a multi-aJne <eld. For the
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sake of simplicity, we neglect intermittency by assuming a pure self-aJne <eld with a unique HVolder exponent h = 13 . Let us
where xn; k is the center of the kth wavelets at the level n, i.e. for eddies with size ‘n ∼ 2−n . According to the Richardson–Kolmogorov cascade picture, one assumes that sweeping is present, i.e., xn+1; k = xn; k + rn+1; k where (n; k ) labels the “mother” of the (n + 1; k)-eddy and rn+1; k is a stochastic vector which depends on rn; k and evolves with characteristic time n ˙ (‘n )1−h . If the coeJcients {an; k } and {rn; k } have characteristic time n ∼ (‘n )1−h and {an; k } ∼ (‘n )h , it is possible to show (see Appendix D for details) that the <eld (5.29) has the correct spatio-temporal statistics, i.e. |v(x + R; t0 ) − v(x; t0 )| ∼ |R|h ;
(5.30)
|v(x; t0 + t) − v(x; t0 )| ∼ t h :
(5.31)
In addition the proper Lagrangian sweeping is satis<ed. Now we are ready for the j-entropy analysis of the <eld (5.29). If one wants to look at the <eld v with a resolution j, one has to take n in (5.29) up to N given by (‘N )h ∼ j ;
(5.32)
in this way one is sure to consider velocity 8uctuations of order j. Then the number of terms contributing to (5.29) is #(j) ∼ (2d )N ∼ j−d=h :
(5.33)
By using a result of Shannon [201] one estimates the j-entropy of the single process an; k (t) (and also of rn; j ) as 1 1 ; (5.34) hn (j) ∼ ln n j where the above relation is rigorous if the processes an; k (t) are Gaussian and with a power spectrum di#erent from zero on a band of frequency ∼ 1=n . The terms which give the main contribution are those with n ∼ N with N ∼ (‘N )1−h ∼ j((1−h)=h) . Their number is given by (5.33) so that, collecting the above results, one
(5.36)
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By denoting with vK the typical velocity at the Kolmogorov scale K, one has that Eq. (5.36) holds in the inertial range, i.e., j ¿ vK ∼ Re−1=4 , while for j 6 vK , hST (j) = constant ∼ Re11=4 . Let us now consider an alternative way to compute the j-entropy of the <eld v(x; t): divide the d-volume in boxes of edge length ‘(j) ∼ j1=h and look at the signals v(x= ; t), where the x= are the centers of the boxes. Denoting with h(=) (j) the j-entropy of the temporal sequence of the velocity <eld measured in x= , we have h(=) (j) ∼ j−1=h
(5.37)
because of the scaling (5.31). Therefore, hST (j) is obtained summing up all the “independent” contributions (5.37), i.e. hST (j) ∼ N(j)h(=) (j) ∼ N(j)j−1=h ;
(5.38)
where N(j) is the number of independent cells. It is easy to understand that the simplest assumption N(j) ∼ l(j)d ∼ jd=h gives a wrong result, indeed one obtains hST (j) ∼ j−(d+1)=h ;
(5.39)
which is not in agreement with (5.35). In order to obtain the correct result (5.36) it is necessary to assume N(j) ∼ l(j)D ;
(5.40)
with D =d − h. In other words, one has that the sweeping implies a non-trivial spatial correlation, quantitatively measured by the exponent D, which can be considered as a sort of “entropy” dimension. Incidentally, we note that D has the same numerical value of the fractal dimensions of the velocity iso-surfaces [154,218]. From this observation, at
where now the xn; k are <xed and no longer time-dependent, while an; k ∼ (‘n )h but n ∼ ‘n . We conclude by noting that it is possible to obtain (see [89]) the scaling (5.35) using Eq. (5.41), i.e. ignoring the sweeping, assuming n ∼ (‘n )1−h and an; k ∼ (‘n )h ; this corresponds to take separately the proper temporal and spatial spectra. However, this is not satisfactory since one has not the proper scaling in one <xed point (see Eq. (5.37) the only way to obtain this is through the sweeping). 6. Uncertainty in the evolution equations The study of a large class of problems in science (physics, chemistry, biology, etc.) is reduced to the investigation of evolution laws, which describe some aspects of the system. The
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assumption that natural processes can be described by mathematical models is at the foundation of this approach [220,70]. The purpose of this section is to discuss how the unavoidable uncertainty in the equation of motion puts limits on the long time forecasting. To be more concrete, let us consider a system described by a di#erential equation d (6.1) x(t) = f(x; t); x; f ∈ Rn : dt As a matter of fact, we do not know exactly the equations, so we have to devise a model which is di#erent from the true dynamics. In practice, this means that we study d (6.2) x(t) = fj (x; t) where fj (x; t) = f(x; t) + jf(x; t) : dt Therefore, it is natural to wonder about the relation between the true evolution (reference or true trajectory xT (t)) given by (6.1) and the one e#ectively computed (perturbed or model trajectory xM (t)) given by (6.2). A typical example is the relation between the true dynamics of the physical system and the one obtained by a computer simulation. This issue is of particular relevance for the study of weather forecast where it is referred to as predictability of the second kind [175]. In this context it is particularly relevant the shadowing lemma [41] which implies that, for Anosov systems, a computer may not calculate the true orbit but what it does
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Fig. 24. The map fj of Eq. (6.3) (solid line) and the original chaotic map f (dashed line).
If one compares the trajectories obtained iterating f(x) or fj (x) it is not diJcult to understand that they may remain identical for a certain time but unavoidably di#er utterly in the long time behavior. The transient chaotic behavior of the perturbed orbits can be rendered arbitrarily long by reducing the interval in which the two dynamics di#er [28]. As for the problem of predictability with respect to perturbations on the initial conditions, the problem of second kind predictability in the limit of in
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Fig. 25. Finite size Lyapunov exponents TT ()(+) and TM ()(×) versus for the Lorenz model (3:29) with 1 = c = 10; b = 83; r = 45 and j = 0:001. The dashed line represents the largest Lyapunov exponent for the unperturbed system (T ≈ 1:2). The statistics is over 104 realizations.
chaotic case (iii) one expects that the perturbed dynamics is still chaotic. In the following we will consider only this latter case. In chaotic systems, the e#ects of a small uncertainty on the evolution law is, for many aspects, similar to those due to imperfect knowledge of initial conditions. As an example let us consider the Lorenz system (3.29). In order to mimic an indetermination in the evolution law we assume a small error j on the parameter r : r → r + j. Let us consider the di#erence x(t) = xM (t) − xT (t), for simplicity, x(0) = 0, i.e. we assume a perfect knowledge of the initial condition. For small j one has, with obvious notation: dx 9f 9fj j: = fj (xM ) − f(xT ) x + dt 9x 9r
(6.4)
Since at time t = 0 one has |x(0)| = 0, |x(t)| initially grows under the e#ect of the second term in (6.4). At later times, when |x(t)| ≈ O(j) the
ln(|x(t)|) follows the usual linear growth with the slope given by the largest LE. Typically the value of the LE computed by using the model dynamics di#ers from the true one by a small amount of order j, i.e. M = T + O(j) [64]. A picture of the error growth, valid also for
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recovers the intrinsic predictability of the true system. For very small errors, TM is dominated by the second term in (6.4) and deviates from TT . 6.1. Uncertainty introduced by numerical computations In numerical computations, an unavoidable source of errors is due to the representation of numbers on the computer, as computers work with integers. This has two main consequences: the phase space of the simulated system is necessarily discrete (and
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Fig. 26. ln |x(t)| versus t, where |x(t)| is the Euclidean distance between two trajectories of the Lorenz model (3.29) for r =28. Curve (a) refers to the comparison between trajectories obtained using a fourth-order Runge–Kutta algorithm with \t = 4 × 10−3 and \t = 4 × 10−5 . Curve (b) shows the same quantity obtained with \t = 4 × 10−4 and \t = 4 × 10−5 . The dotted line with slope ≈ 0:9 is shown for comparison.
be explicitly resolved. In this framework, a natural question is: how must we parameterize the unresolved modes in order to predict the resolved ones? In this respect, the optimal parameterization is that one for which the predictability on the resolved modes is not worse than the intrinsic predictability of the same variables in the complete system, i.e. in our notation TM = TT . An example in which it is relatively simple to develop a model for the small-scale modes is represented by skew systems, i.e., g depends only on the fast variables y. In this case, simply neglecting the fast variables or parameterizing them with a suitable stochastic process does not drastically a#ect the prediction of the slow variables [32]. On the other hand, in typical cases y feels some feedback from x, and, therefore, we cannot simply neglect the unresolved modes (see Ref. [36] for details). In practice, one has to construct an e#ective equation for the resolved variables: dx (6.6) = fM (x; y(x)) ; dt where the functional form of y(x) and fM is built by phenomenological arguments and=or by numerical studies of the full dynamics (if available). Let us now discuss a simpli<ed model for atmosphere circulation [147,148] which includes large scales xk (synoptic scales) and small scales yj; k (convective scales): J d xk yj; k ; = −xk−1 (xk−2 − xk+1 ) − Cxk + F − dt j=1
dyj; k (6.7) = −cbyj+1; k (yj+2; k − yj−1; k ) − cCyj; k + xk ; dt where k = 1; : : : ; K and j = 1; : : : ; J . As in [147] we assume periodic boundary conditions on k (xK+k = xk , yj; K+k = yj; k ) while for j we impose yJ +j; k = yj; k+1 . The variables xk represent some
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Fig. 27. Finite size Lyapunov exponents for the Lorenz model (6.7) TT () (solid line) and TM () versus obtained by dropping the fast modes (+) and with eddy viscosity parameterization (×). The parameters are F = 10; K = 36; J = 10; C = 1 and c = b = 10, implying that the typical y variable is 10 times faster and smaller than the x variabe. The value of the parameter Ce = 4 is chosen after a numerical integration of the complete equations as discussed in Ref. [36]. The statistics is over 104 realizations.
large scale atmospheric quantities in K sectors extending on a latitude circle, while the yj; k represent quantities on smaller scales in J · K sectors. The parameter c is the ratio between fast and slow characteristic times and b measures the relative amplitude (both larger than unity). Model (6.7), even if rather crude, contains some nontrivial aspects of the general circulation problem, namely the coupling among variables with very di#erent characteristic times. Being interested in forecasting the large scale behavior of the atmosphere by using only the slow variables, a natural choice for the model equations is: d xk = −xk−1 (xk−2 − xk+1 ) − Cxk + F − Gk (x) ; dt
(6.8)
where Gk (x) represents the parameterization of the fast components in (6.7). Following the approach discussed in Ref. [36], a physical reasonable parameterization is Gk (x) = Ce xk ;
(6.9)
where Ce is a numerically determined parameter. In Fig. 27 we plot TM () obtained from di#erent choices of Gk . The simplest possibility is to neglect the fast variable, i.e. Gk =0. Also for large errors we have TM () ¿ TT () because this crude approximation is not able to capture the intrinsic predictability of the original system. More re
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Fig. 28. The FSLE for the eddy-viscosity shell model (6.11), (6.12) MM () at various resolutions NM = 9(+); 15(×); 20(∗). For comparison it is drawn TT () (continuous line). Here M = 0:4; k0 = 0:05.
As a more complex example, let us consider a version of the shell model discussed in Section 5.2, more precisely we study [151] dun ∗ ∗ un+2 − 12 kn un−1 un+1 + 12 kn−1 un−2 un−1 ) − Ckn2 un + fn ; (6.10) = i(kn+1 un+1 dt with n = 1; : : : ; N . At variance with the previous example, here we have a set of scales ‘n 1=kn with characteristic times n ∼ kn−2=3 (see Section 5.4). In order to simulate a
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Fig. 29. The FSLE between the eddy-viscosity shell model and the full shell model TM (), at various resolutions NM = 9(+); 15(×); 20(∗). For comparison it is drawn TT () (continuous line). The total number of shell for the complete model is N = 24, with k0 = 0:05; C = 10−7 .
resolution NM = 9; 15; 20 towards the fully resolved case N = 24 the model improves, in agreement with the expectation that TM approaches TT for a perfect model. At large the curves practically coincide, showing that the predictability time for large error sizes (associated with large scales) is independent of the details of small-scale modeling. 6.3. Lyapunov exponents and complexity in dynamical systems with noise We saw how in deterministic dynamical systems there exist well established ways to de
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We will show how, for noisy and random systems, a more natural indicator of complexity can be obtained by computing the separation rate of nearby trajectories evolving with di#erent noise realizations. This measure of complexity, de
1 1 t 2 9 V (x(t )) dt 1 = lim ln |z(t)| = − lim t→∞ t t→∞ t 0 xx
C 2 −V (x)=1 = −C 9xx V (x)e dx = − (9x V (x))2 e−V (x)=1 d x ¡ 0 : (6.16) 1 This result has a rather intuitive meaning: the trajectory x(t) spends most of the time in one of the “valleys” where −92xx V (x) ¡ 0 and only short intervals on the “hills” where −92xx V (x) ¿ 0, so that the distance between two trajectories evolving with the same noise realization decreases on average. Notice that in Ref. [215], supported by a wrong argument, an opposite conclusion has been claimed. A negative value of 1 implies a fully predictable process only if the realization of the noise is known. In the case of two initially close trajectories evolving under two di#erent noise realizations, after a certain time T1 , the two trajectories can be very distant, because they can be in two di#erent valleys. For 1 → 0, due to the Kramers formula [57], one has T1 ∼ exp \V=1, where \V is the di#erence between the values of V on the top of the hill and at the bottom of the valley. The result obtained for the one dimensional Langevin equation can easily be generalized to any dimension for gradient systems if the noise is small enough [149]. Another example showing the limitations of this approach is provided by the case of stochastic resonance in chaotic systems. In this case, in fact, one can
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6.3.2. An information theory approach The main diJculties in de
(6.17)
where t is an integer and w(t) is an uncorrelated random process, e.g. w are independent random variables uniformly distributed in [ − 12 ; 12 ]. For the largest LE 1 , as de
(6.18)
where f = df=d x. Following the approach of Section 2.3, let x(t) be the trajectory starting at x(0) and x (t) be the trajectory starting from x (0) = x(0) + x(0). Let 0 ≡ |x(0)| and indicate by 1 the minimum time such that |x (1 ) − x(1 )| ¿ . Then, we put x (1 ) = x(1 ) + x(0) and de
(6.21)
The interesting situation happens for strong intermittency when there are alternations of positive and negative + during long time intervals: this induces a dramatic change for the value of K1 . This becomes particularly clear when we consider the limiting case of positive +(1) in an interval T1 1=+(1) followed by a negative +(2) in an interval T2 1= |+(2) |, and again a positive e#ective LE and so on. During the intervals with positive e#ective LE the transmission has to be repeated rather often with T1 =(+(1) ln 2) bits at each time, while during the ones with negative e#ective LE no information has to be sent. Nevertheless, at the end of the contracting intervals one has |x| = O(1), so that, at variance with the noiseless case, it is impossible to use them to compensate the expanding ones. This implies that in the limit of very large Ti only the
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Fig. 30. K1 versus T with 1 = 10−7 for the map (6.23). The parameters of the map are a = 2 and b = or b = 14 (circles). The dashed lines are the noiseless limit of K1 .
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2 3
(squares)
expanding intervals contribute to the evolution of the error x(t) and K1 is given by an average of the positive e#ective Lyapunov exponents: K1 +5(+) :
(6.22)
Note that it may happen that K1 ¿ 0 with 1 ¡ 0. We stress again that (6.22) holds only for strong intermittency, while for uniformly expanding systems or rapid alternations of contracting and expanding behaviors K1 1 . Note that K1 is a sort of j-entropy (see Section 3.5), indeed, the complexity we consider is de
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Fig. 31. 1 (squares) and K1 (crosses) versus 1 for the map (6.24).
The second example (Fig. 31), strongly intermittent without external forcing, is the Beluzov– Zhabotinsky map [103,157], introduced for describing the famous chemical reaction: [(1=8 − x)1=3 + a]e−x + b if 0 6 x ¡ 1=8 ; (6.24) f(x) = [(x − 1=8)1=3 + a]e−x + b if 1=8 6 x ¡ 3=10 ; c(10xe−10x=3 )19 + b if 3=10 6 x ; with a=0:50607357; b=0:0232885279; c=0:121205692. The map exhibits a chaotic alternation of expanding and contracting time intervals. In Fig. 31, one sees that while 1 passes from negative to positive values at decreasing 1, K1 is not sensitive to this transition [157]. Considering the system with a given realization of noise as the “true” evolution law, one has that 1 corresponds to TT while K1 corresponds to TM . The previous results show that the same system can be regarded either as regular (i.e. 1 ¡ 0), when the same noise realization is considered for two nearby trajectories, or as chaotic (i.e. K1 ¿ 0), when two di#erent noise realizations are considered. 6.4. Random dynamical systems We discuss now dynamical systems where the randomness is not simply given by an additive noise. This kind of systems has been the subject of interest in the last few years in relation to the problems involving disorder [119], such as the characterization of the so-called on–o5 intermittency [187] and to model transport problems in turbulent 8ows [227,228,86]. In these systems, in general, the random part represents an ensemble of hidden variables believed to be implicated in the dynamics. Random maps exhibit very interesting features ranging from stable or quasi-stable behaviors, to chaotic behaviors and intermittency. In particular, on–o5 intermittency is an aperiodic switching between static, or laminar, behavior and chaotic bursts of oscillation. It can be generated by systems having an unstable invariant manifold, within
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which it is possible to
(6.25)
where J (t) is a random variable. As for the case of additive noise examined in the previous section, the simplest approach is the introduction of the LE J computed considering the separation of two nearby trajectories evolving with the same realization of the random process J (t) = i1 ; i2 ; : : : ; it . The Lyapunov exponent J generalizes 1 of Section 6.3.1 and can be computed from the tangent vector evolution: 1 J = lim (6.26) ln |z(N )| ; N →∞ N where 9fm (x(t); it ) zn (t) : (6.27) zm (t + 1) = 9x n n On the other hand, also for these systems, as in the case of additive noise, it is possible to introduce a measure of complexity, K, which better accounts for their chaotic properties [174,149] K hSh + J 5(J ) ;
(6.28)
where hSh is the Shannon entropy of the random sequence J (t). The meaning of K is rather clear: K=ln 2 is the mean number of bits, for each iteration, necessary to specify the sequence x1 ; : : : ; xt with a certain tolerance . Note that there are two di#erent contributions to the complexity: (a) one has to specify the sequence J (1); J (2); : : : ; J (t) which implies hSh =ln 2 bits per iteration; (b) if J is positive, one has to specify the initial condition x(0) with a precision \ exp−J T , where T is the time length of the evolution. This requires J =ln 2 bits per iteration; if J is negative the initial condition can be speci<ed using a number of bits independent of T . 6.4.1. A toy model: one-dimensional random maps Let us discuss a random map which, in spite of its simplicity, captures some basic features of this kind of systems [187,101]: x(t + 1) = at x(t)(1 − x(t)) ; where at is a random dichotomous variable given by 4 with probability p ; at = 1=2 with probability 1 − p : For x(t) close to zero, we can neglect the non-linear term to obtain t−1 x(t) = aj x(0) ; j=0
(6.29)
(6.30)
(6.31)
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Fig. 32. x(t) versus t for the random map (6.29), (6.30), with p = 0:35.
from the law of large numbers one has that the typical behavior is x(t) ∼ x(0)e ln a t :
(6.32)
Since ln a = p ln 4 + (1 − p) ln 1=2 = (3p − 1) ln 2 one has that, for p ¡ pc = 1=3, x(t) → 0 for t → ∞. On the contrary for p ¿ pc after a certain time x(t) escapes from the <xed point zero and the non-linear term becomes relevant. Fig. 32 shows a typical on–o5 intermittency behavior for p slightly larger than pc . Note that, in spite of this irregular behavior, numerical computations show that the LE J is negative for p ¡ p˜ c 0:5: this is essentially due to the non-linear terms. By introducing a
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of the self-organized criticality (SOC) [14]. This term refers to the tendency of some large dynamical systems to evolve spontaneously toward a critical state characterized by spatial and temporal self-similarity. The original sandpile models are probabilistic cellular automata inspired to the dynamics of avalanches in a pile of sand. Dropping sand slowly, grain by grain on a limited base, one reaches a situation where the pile is critical, i.e. it has a critical slope. This means that a further addition of sand will produce sliding of sand (avalanches) that can be small or cover the entire size of the system. In this case the critical state is characterized by scale-invariant distributions for the size and the lifetime and it is reached without tuning of any critical parameter. We will refer in particular to the Zhang model [229], a continuous version of the original sandpile model [15], de
(6.34)
where nn indicates the 2d nearest neighbors of the site i; (c) one repeats point (b) until all the sites are relaxed; (d) one goes back to point (a). Let us now discuss the problem of predictability in sandpile models on the basis of the rigorous results [46], which clarify the role of the LE for this class of systems. In Ref. [46] it has been proved that the LE J is negative. In fact the dynamics of a little di#erence between two con
const ; R2
(6.35)
where R is the diameter of the system. As for other examples already discussed, the existence of a negative LE does not mean a perfect predictability. This can be understood by looking at the growth of the distance, (t), between two initially close trajectories computed with two di#erent realizations of randomness, i.e., by adding sand in di#erent sites. Let us consider the case of the “minimal error”: in the reference realization one adds sand on a site i chosen at random. In the perturbed realization, instead, one adds a sand grain at one of the nearest sites of i. In such a case (t) increases up to a maximal distance in few avalanches [150]. Practically, one has the same kind of phenomenon, already discussed, of the Langevin equation with two noise realizations. Let us now estimate the complexity K of this system. An upper bound can be given by using (6.28) K = hSh + J 5(J ), where hSh is the entropy of the random sequence of addition
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Fig. 33. Schematic representation of the evolution of a deterministic rule with a
of energy. In sandpile models, since each site has the same probability to be selected, one has hSh = ln V , where V is the number of sites of the system. Since the Lyapunov exponent is negative, the complexity is just determined by hSh . 7. Irregular behavior in discrete dynamical systems For the sake of completeness we include in this review a discussion on the characterization of irregular behaviors in systems whose states are discrete. Such systems include Cellular Automata (CA), which have been intensively studied both for their intrinsic interest [223] and for applications as, e.g., to simulate hydrodynamic equations [82] or to study various forms of chemical turbulence [166,167,40]. Other interesting systems with discrete states are the neural networks used for modeling some brain functions [6]. It is also relevant to note that in every simulation with a computer, because of the
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The above conclusions, although mathematically correct, are rather unsatisfactory from the physical point of view, indeed from this side the following questions deserve some interest: (1) What is the “typical” period, T, in a system with N elements, each assuming k distinct values? (2) When T is very large, how can we characterize the (possible) irregular behavior of the trajectories, on times that are large enough but still much smaller than T? (3) What does it happen in the transition from discrete to continuous states, i.e. in the limit k → ∞? In the next subsections we will deal with the above questions. 7.1. Dependence of the period on the number of the states For deterministic discrete state systems the dependence of the period of the attractor on the number of the states, may be addressed with a statistical approach in terms of random maps [63]. We recall that this problem is important for computer simulations of chaotic systems (see Section 6.1). If N = k N 1 is the number of states of the system, the basic result for the average period, T, is √ T(N) ∼ N : (7.1) In the following we give a simple argument, by Coste and HKenon [63]. For simplicity of notation, we consider the case with k = 2, so that the state of the system is a string of N bits. A deterministic evolution of such a system is given by a map which is one among the possible functions connecting the 2N states. Let us now assume that all the possible functions can be extracted with the same probability. Denoting by I (t) the state of the system, for a certain map we have a periodic attractor of period m if I (p + m) = I (p) and I (p + j) = I (p), for j ¡ m. The probability, !(m), of this periodic orbit is obtained by specifying that the (p + m − 1)th
e−m =2N M (m) = !(m) ≈ ; (7.3) m m √ from which one obtains T ∼ N for the average period. It is here appropriate to comment on the relevance of Eq. (7.1) for computer-generated orbits of chaotic dynamical systems. Because of the
(N1)
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[93,94], one can reasonably assume that N ∼ 10nd2 , so that, from Eq. (7.1) one has T ∼ 10nd2 =2 :
(7.4)
This estimation gives an upper limit for the typical number of meaningful iterations of a map on a computer. Note that this number, apart from the cases of 1 or 2-dimensional maps with few digits, is very large for almost all practical purposes. 7.2. The transition from discrete to continuous states Following the basic ideas of Ford [79,80], as discussed in Section 2.3, and the results of Section 6—on the predictability in systems whose evolution law is not completely known—we describe now a way to introduce a practical de
(7.5)
A single state I is a sequence of (at most) ln2 N bits, and its time evolution for M steps can be surely translated in a binary sequence O of length ‘O (M; N) 6 M ln2 N. Relying one the de
(7.6)
Let us note that from the above equation one has that – when M grows inde
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initial conditions I(0). Let us consider C(2) as an j-perturbation of C(1) , i.e. we pose, for each component of the parameter vector: Ci(2) = Ci(1) + ji ;
(7.7)
where the random variables ji are uniformly distributed in [ − 2(−q−1) ; 2(−q−1) ]. Let us note that the coding length O(Cq) + ln2 N is enough to de
M˜ (j) ∼ (7.9) for j ¡ jc (L) : 1= j It is rather easy to give analytical estimates supporting the numerical evidence [65] .l for j ¡ l ; j
M˜ (j) = −1 25 5 j l for j ¿ l : (1 − cos 5) + 1 − + ln tan .l . .j 2
(7.10)
where the angle 5 is de
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farther than a <xed distance in the discrete phase space of Eq. (7.8). We found (1=) ln(= j) for j ¿ jc (L) ;
M (j) ∼ for j ¡ jc (L) : 1= j
(7.11)
We remark that when j ¡ jc (L), M (j) is weakly dependent, i.e. logarithmically, on . This is just another veri
i = 1; : : : ; N ;
(7.12)
where r de
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Fig. 34. Typical behavior of (a) regular [Rule 52], (b) chaotic [Rule 22], (c) complex [Rule 20]. We used totalistic r = 2 cellular automata. Time 8ows from below to above.
speaks of “totalistic” CA. Another usual requirement is to have symmetric rules. For further details we refer to Ref. [223], where the standard scheme for the classi
(7.13)
Moreover, also the cycle period shows in most of the cases a similar dependence on N , this is a reminiscence of what we discussed in Section 7.1.
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Complex cellular automata (class 4 in [223], Fig. 34c) usually evolve toward complex localized structures (gliders) which interact in a complicate way. For these CA numerical simulations [100] have shown that both the transient time and the cycle period display a non-trivial N -dependence (i.e. the average, the typical values or the median depend in a di#erent way on N ). The unpredictability of these system manifests itself in the distribution of these times. In particular, the large variability of these times in dependence of the initial conditions and the lattice size inhibits any forecasting of the duration of the transient. In the following we limit the discussion to chaotic rules, i.e. class 3 in the Wolfram classi
where With the above norm two systems can be arbitrarily close: one only needs N0 1. At this point it is possible to de
(7.17)
and therefore = v ln 2 :
(7.18)
In other words, the linear damage spreading in the physical space corresponds to an exponential growth in the norm (7.15). Oono and Yeung [167] stressed a conceptual (and practical) diJculty with the above approach. In systems with continuous states it is clear that by performing an
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in
91i (t + 1) ≡ F[1i−1 ; 1i ; 1i+1 ] 91i−1 (t)
XOR
F[1i−1
XOR
1; 1i ; 1i+1 ]
where the other non-zero terms are obtained by shifting the XOR operation to i and i + 1 (respectively). We recall that XOR is the Boolean exclusive operation (i.e. 0 XOR 0=0; 1 XOR 1=0, 0 XOR 1 = 1 and 1 XOR 0 = 1). Of course as time goes on the initial perturbation spreads, i.e. new defects appear. As in continuous systems, one needs to maintain the perturbation “in
Finally, putting |N(t)| = automaton as B = lim
T →∞
j
Nj (t), one can de
1 ln (|N(T )|) : T
(7.20)
Now, in analogy with continuous systems, B ¡ 0 indicates an exponential decrease of the perturbation, while for B ¿ 0 the damage spreads. Just to give an example, if one considers the rule 150 of Wolfram classi
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Fig. 35. Damage spreading analysis performed on a totalistic [Rule 10] r = 2 cellular automaton with N = 200. At time t = 0 a replica is initialized by 8ipping the value at the center of the lattice.
7.3.3. Entropies For cellular automata one can de
and then the spatio-temporal entropy density as 1 hST = lim h(L) : (7.22) L→∞ L This entropy cannot be practically computed. A more accessible quantity is the temporal entropy: 1 P(C(1; T )) ln P(C(1; T )) ; (7.23) hT = h(1) = lim − T →∞ T C(1;T )
i.e. the Shannon entropy of the time sequence of one element (1n (0); 1n (1); : : :). In principle, hT can depend on the site n and one can classify as non-trivial a system for which the majority of the elements have hT ¿ 0 [166]. An average measure of the “temporal disorder” is given by the spatial average hT . A systematic study of h(1); h(2); h(3); : : : – although very diJcult in practice – could give, in principle, relevant information on the spatial=temporal behavior. A characterization of the spatial properties can be obtained studying, at a given time t, the spatial sequences. In practice, one studies C(L; 1) at increasing L: 1 hS = lim − P(C(L; 1)) ln P(C(L; 1)) : (7.24) L→∞ L C(L;1)
One can associate to hS a sort of “e#ective” dimension d = hS =ln 2 [223]. In a completely disordered cellular automaton con
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Fig. 36. Sketch of the dependence of temporal sequences on spatial ones. Fig. 37. The values of the sites in black together with the speci
From the de
(7.25)
where a good estimate of vp can be given in terms of the damage spreading velocity (7.18) [223]. The possible scenario arising from (7.25) can be summarized as follows. One can have “spatial chaos” (hS ¿ 0) in absence of “temporal chaos” (hT = 0), while the existence of “temporal chaos” requires not only a non-zero spatial entropy but also the existence of a
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indeed seems to be the natural candidate for such a comparison. We limit the discussion to 1-dimensional lattices with r = 1, i.e. CML and CA with nearest neighbor coupling. Let us now ask the amount of information we have to specify for knowing all the LT sites of spatial size L(¡ N ) and temporal length T , as shown in Fig. 37. Since both CA and CML are ruled by a local deterministic dynamics one needs to specify the rule of evolution and the values of the L + 2(T − 1) states at the border, in black in Fig. 37. Basically, one has to specify the initial conditions on the L sites and the “boundaries” 11 (t) and 1L (t) for 1 ¡ t 6 T . But while for CA this speci
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the diJculty in distinguishing, only from data analysis, a genuine deterministic chaotic system from one with intrinsic randomness [51]. On the other hand, the way the j-entropy or
(8.1)
with an integer multiplier N1 and modulus N2 . The {zn } are integer numbers and one hopes to generate a sequence of random variables {x n }, which are uncorrelated and uniformly distributed in the unit interval. A
(8.2)
which has a uniform invariant measure and a KS entropy hKS = = ln N1 . When imposing the integer arithmetics of Eq. (8.1) onto this system, we are, in the language of dynamical systems, considering an unstable periodic orbit of Eq. (8.2), with the particular constraint that, in order to achieve the period N2 − 1 (i.e. all integers ¡ N2 should belong to the orbit of Eq. (8.1)) it has to contain all values k=N2 , with k = 1; 2; : : : ; N2 − 1. Since the natural invariant measure of Eq. (8.2) is uniform, such an orbit represents the measure of a chaotic solution in an optimal way. Every sequence of a PRNG is characterized by two quantities: its period T and its positive Lyapunov exponent , which is identical to the entropy of a chaotic orbit of the equivalent dynamical system. Of course a good random number generator has a very large period, and as large as possible entropy. It is natural to ask how this apparent randomness can be reconciled with the facts that (a) the PRNG is a deterministic dynamical systems (b) it is a discrete state system. If the period is long enough on shorter times one has to face only point (a). In the following we discuss this point in terms of the behavior of the j-entropy, h(j) (see Section 3.5). It seems rather reasonable to think that at a high resolution, i.e. j 6 1=N1 , one should realize the true deterministic chaotic nature of the system and, therefore, h(j) hKS = ln N1 . On the other hand for j ¿ 1=N1 one expects to observe the “apparent random” behavior of the system, i.e. h(j) ∼ ln (1= j). When the spatial resolution is high enough so that every point of this periodic orbit is characterized by its own symbol, then, for arbitrary block length m, one has a
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Fig. 38. The j-entropies, hm (j), at varying the embedding dimension m for the multiplicative congruential random number generator, Eq. (8.1), for di#erent choices of N1 and N2 . This
of m-words whose probabilities are di#erent from 0. Therefore, the block entropy Hm (3.44) is m-independent and hm = 0. In Fig. 38 it is shown the behavior of hm (j), computed on sequences of length 60000 of the PRNG with three di#erent pairs (N1 ; N2 ) chosen to be (75 ; 232 ), (2; 494539), and (107; 31771). The
(8.3)
where j = 1; : : : ; N denotes the site of a lattice of size N , t the discrete time and 1 the coupling strength. A detailed numerical study (also supported by analytical arguments) of the j-entropy hm (j) at di#erent j, in the limit of small coupling, gives the following scale-dependent scenario: for 1 ¿ j ¿ 1 there is a plateau h(j) s where s is the Lyapunov exponent of the single map x(t + 1) = f(x(t)). For 1 ¿ j ¿ 12 another plateau appears at h(j) 2s , and so on: for 1n−1 ¿ j ¿ 1n one has h(j) ns (see Fig. 39). Similar results hold for the correlation dimension which increases step by step as the resolution increases, showing that the high-dimensionality of the system becomes evident only as j → 0.
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Fig. 39. hm (j) for the system (8.3) where f(x) = 2|1=2 − |x − 1=2 is the tent map and 1 = 0:01. The horizontal lines indicate the entropy steps which appears at decreasing j. The j-entropy is computed with the Grassberger–Procaccia method [92]. For further details see Ref. [165].
Therefore one understands that the dynamics at di#erent scales is basically ruled by a hierarchy of low-dimensional systems whose “e#ective” dimension ne# (j) increases as j decreases [165]: ln (1= j) ; (8.4) ne# (j) ∼ ln (1=1) where [ : : : ] indicates the integer part. In addition, for a given resolution j, it is possible to
1 j
(8.5)
i.e. the typical behavior of a stochastic process. Of course, for j 6 1N one has to realize that the system is deterministic and h(j) = O(Ns ). Let us now brie8y reconsider the issue of the macroscopic chaos, discussed in Section 4.7. The main result can be summarized as follows: √ • at small j (1= N ), where N is the number of elements, one recovers the “microscopic” 2 , i.e. (j) ≈ Lyapunov exponent micro √ • at large j (1= N ) one observes another plateau (j) ≈ macro which can be much smaller than the microscopic one. 2
From hereafter we use the same symbol j both for the FSLE and the j-entropy in order to make a direct comparison between the two quantities.
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√ The emerging scenario is that at a coarse-grained level, i.e. j1= N , the system can be described by an “e#ective” hydro-dynamical equation (which in some cases can be lowdimensional), while the “true” high-dimensional character appears only at very high resolution, i.e. 1 j 6 jc = O √ : N
8.3. Di5usion in deterministic systems and Brownian motion Consider the following map which generates a di#usive behavior on the large scales [200]: xt+1 = [xt ] + F(xt − [xt ]) ; where [xt ] indicates the integer part of xt and F(y) is given by: (2 + =)y if y ∈ [0; 1=2] ; F(y) = (2 + =)y − (1 + =) if y ∈ [1=2; 1] :
(8.6)
(8.7)
The largest Lyapunov exponent can be obtained immediately: =ln |F |, with F =dF=dy=2+=. One expects the following scenario for h(j): h(j) ≈
for j ¡ 1 ;
D for j ¿ 1 ; j2 where D is the di#usion coeJcient, i.e. h(j) ˙
(xt − x0 )2 ≈ 2Dt
for large t :
(8.8) (8.9)
(8.10)
Consider now a stochastic system, namely a noisy map xt+1 = [xt ] + G(xt − [xt ]) + 1Kt ;
(8.11)
where G(y), as shown in Fig. 40, is a piecewise linear map which approximates the map F(y), and Kt is a stochastic process uniformly distributed in the interval [ − 1; 1], and no correlation in time. When |dG=dy| ¡ 1, as is the case we consider, the map (8.11), in the absence of noise, gives a non-chaotic time evolution. Now we compare the
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Fig. 40. The map F(x) (8.7) for = = 0:4 is shown with superimposed the approximating (regular) map G(x) (8.11) obtained by using 40 intervals of slope 0.
Fig. 41. (j) versus j obtained with the map F(y) (8.7) with = = 0:4 (◦) and with the noisy (regular) map ( ) (8.11) with 10 000 intervals of slope 0.9 and 1 = 10−4 . The straight lines indicates the Lyapunov exponent = ln 2:4 and the di#usive behavior (j) ∼ j−2 .
and larger than zero. Finally, on the smallest length scales j ¡ 1 we see stochastic behavior for the system (8.11) while the system (8.6) still shows chaotic behavior. 8.4. On the distinction between chaos and noise The above examples show that the distinction between chaos and noise can be a high non-trivial task, which makes sense only in very peculiar cases, e.g., very low-dimensional systems. Nevertheless, even in this case, the entropic analysis can be unable to recognize the “true” character of the system due to the lack of resolution. Again, the comparison between the
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di#usive map (8.6) and the noisy map (8.11) is an example of these diJculties. For 1 6 j 6 1 both the system (8.6) and (8.11), in spite of their “true” character, will be classi<ed as chaotic, while for j ¿ 1 both can be considered as stochastic. In high-dimensional chaotic systems, with N degrees of freedom, one has typically h(j) = hKS ∼ O(N ) for j 6 jc (where jc → 0 as N → ∞) while for j ¿ jc , h(j) decreases, often with a power law [89]. Since also in some stochastic processes the j-entropy obeys a power law, this can be a source of confusion. These kind of problems are not abstract ones, as a recent debate on “microscopic chaos” demonstrates [90,72,98]. The detection of microscopic chaos by data analysis has been recently addressed in a work of Gaspard et al. [90]. These authors, from an entropic analysis of an ingenious experiment on the position of a Brownian particle in a liquid, claim to give an empirical evidence for microscopic chaos. In other words, they state that the di#usive behavior observed for a Brownian particle is the consequence of chaos at a molecular level. Their work can be brie8y summarized as follows: from a long (≈ 1:5 × 105 data) record of the position of a Brownian particle they compute the j-entropy with the Cohen–Procaccia method [61] (Section 3) from which they obtain h(j) ∼
D j2
;
(8.12)
where D is the di#usion coeJcient. Then, assuming that the system is deterministic, and making use of the inequality h(j ¿ 0) 6 hKS , they conclude that the system is chaotic. However, their result does not give a direct evidence that the system is deterministic and chaotic. Indeed, the power law (8.12) can be produced with di#erent mechanisms: (1) a genuine chaotic system with di#usive behavior, as the map (8.7); (2) a nonchaotic system with some noise, as the map (8.11), or a genuine Brownian system; (3) a deterministic linear non-chaotic system with many degrees of freedom (see for instance [158]); (4) a “complicated” non-chaotic system as the Ehrenfest wind-tree model where a particle di#uses in a plane due to collisions with randomly placed, <xed oriented square scatters, as discussed by Cohen et al. [72] in their comment to Ref. [90]. It seems to us that the weak points of the analysis in Ref. [90] are: (a) the explicit assumption that the system is deterministic; (b) the limited number of data points and therefore limitations in both resolution and block length. The point (a) is crucial, without this assumption (even with an enormous data set) it is not possible to distinguish between (1) and (2). One has to say that in the cases (3) and (4) at least in principle it is possible to understand that the systems are “trivial” (i.e. not chaotic) but for this one has to use a huge number of data. For example, Cohen et al. [72] estimated that in order to distinguish between (1) and (4) using realistic parameters of a typical liquid, the number of data points required has to be at least ∼ 1034 .
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Concluding, we have the apparently paradoxical result that “complexity” helps in the construction of models. Basically, in the case in which one has a variety of behaviors at varying the scale resolution, there is a certain freedom on the choice of the model to adopt. In Section 8.3 one can see that, for some systems, the behavior at large scales can be realized both with chaotic deterministic models or suitable stochastic processes. From a pragmatic point of view, the fact that in certain stochastic processes h(j) ∼ j−= can be indeed extremely useful for modeling such high-dimensional systems. Perhaps, the most relevant case in which one can use this freedom in modeling is the fully developed turbulence whose non-in
9. Concluding remarks The guideline of this review has been how to interpret the di#erent aspects of the predictability of a system as a way to characterize its complexity. We have discussed the relation between the Kolmogorov–Sinai entropy and the algorithmic complexity (Section 2). As clearly exposed in the seminal works of Alekseev and Yakobson [5] and Ford [79,80], the time sequences generated by a system with sensitive dependence on initial conditions have non-zero algorithmic complexity. A relation exists between the maximal compression of a sequence and its KS-entropy. Therefore, one can give a de
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The study of these di#erent aspects of predictability constitutes a useful method for a quantitative characterization of “complexity”, suggesting the following equivalences:
The above point of view, based on dynamical systems and information theory, quanti<es the complexity of a sequence considering each symbol relevant but it does not capture the structural level. Let us clarify this point with the following example. A binary sequence obtained with a coin tossing is, from the point of view adopted in this review, complex since it cannot be compressed (i.e. it is unpredictable). On the other hand such a sequence is somehow trivial, i.e. with low “organizational” complexity. It would be important to introduce a quantitative measure of this intuitive idea. The progresses of the research on this intriguing and diJcult issue are still rather slow. We just mention some of the most promising proposals as the logical depth [21] and the sophistication [130], see Ref. [13]. As a
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We thank M. Abel, K.H. Andersen, A. Baldassarri, E. Aurell, F. Bagnoli, R. Baviera, A. Bigazzi, R. Benzi, L. Biferale, M. Casartelli, P. Castiglione, A. Crisanti, D.P. Eckmann, U. Frisch, P. Grassberger, K. Kaneko, H. Kantz, M.H. Jensen, G. Lacorata, R. Livi, V. Loreto, R. Mantegna, G. Mantica, U. Marini Bettolo Marconi, A. Mazzino, P. MuratoreGinanneschi, E. Olbrich, G. Parisi, R. Pasmanter, M. Pasquini, A. Pikovsky, A. Politi, A. Provenzale, A. Puglisi, S. Ru#o, M. Serva, A. Torcini, M. Vergassola, D. Vergni and E. Zambianchi for collaborations, correspondence and discussions in the last years. We are grateful to F. Benatti, P. Castiglione, A. Politi and A. Torcini for useful remarks and detailed comments on the manuscript. We acknowledge F. di Carmine, R.T. Lampioni, B. Marani and I. Poeta for their friendly and warm encouragement. Appendix A. On the computation of the 1nite size Lyapunov exponent This appendix is devoted to the methods for computing the
+i i 1 T 1 (n ) = +(n ; r) t = + dt = i = ln r ; (A.2) T 0
( n ; r) e i i where (n ; r) e = i = N and T = i i .
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To obtain Eq. (A.2) we assumed the distance between the two trajectories to be continuous in time. This is not true for maps or for discrete sampling in time and the method has to be slightly modi<ed. In this case the doubling time, (n ; r), is de
(n ; r) e
((n ; r)) ln n
e
:
(A.3)
Let us stress some points. The computation of the FSLE is not more expensive than the one of the Lyapunov exponent by standard algorithm. One has simply to integrate two copies of the system (or two di#erent systems for second kind predictability) and this can be done without particular problems. At di#erence with , () may also depend on the norm one chooses. This fact, apparently disturbing, is however physically reasonable: when one looks at the non-linear regime, for instance, for the predictability problem the answer may depend on the observable considered. A similar problem appears in in
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average of the one-step exponential divergence: 1 x(t + \t) ln ; () = \t x(t) t
461
(A.4)
which, if non-negative, is equivalent to the de
(B.1)
if x ∈ Sh , and Sh is a fractal set with dimension D(h) and h ∈ (hmin ; hmax ). The probability to observe a given scaling exponent h at the scale ‘ is thus P‘ (h) ∼ ‘3−D(h) , so the scaling of the structure function assumes the form
hmax ‘hp ‘3−D(h) dh ∼ ‘Fp ; (B.2) Sp (‘) = v‘p ∼ hmin
where in the last equality, being ‘1, a steepest descent estimation gives Fp = min {hp + 3 − D(h)} = h∗p + 3 − D(h∗ ) ; h
(B.3)
where h∗ = h∗ (p) is the solution of the equation D (h∗ (p)) = p. The Kolmogorov “ 45 ” law [84] S3 (‘) = − 45 j‘
(B.4)
imposes F3 = 1 which implies that 3h + 2 − D(h) 6 0 ;
(B.5) h∗ (3).
the equality is realized by The Kolmogorov similarity theory corresponds to the case of only one singularity exponent h = 13 with D(h = 13 ) = 3. A non-trivial consequence of the intermittency in the turbulent cascade is the 8uctuations of the dissipative scale which implies the existence of an intermediate region between the inertial
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and dissipative range [83]. The local dissipative scale ‘D is determined by imposing the e#ective Reynolds number to be of order unity: vD ‘D Re(‘D ) = ∼1; (B.6) C therefore the dependence of ‘D on h is thus ‘D (h) ∼ LRe−1=(1+h)
(B.7)
where Re = Re(L) is the large-scale Reynolds number. The 8uctuations of ‘D modi<es the evaluation of the structure functions (B.2): for a given ‘, the saddle-point evaluation of (B.2) remains unchanged if, for the selected exponent h∗ (p), one has ‘D (h∗ (p)) ¡ ‘. If, on the contrary, the selected exponent is such that ‘D (h∗ (p)) ¿ ‘ the saddle-point evaluation is not consistent, because at scale ‘ the power-law scaling (B.1) is no longer valid. In this intermediate dissipation range [83] the integral in (B.2) is dominated by the smallest acceptable scaling exponent h(‘) given by inverting (B.7), and the structure function of order p a pseudo-algebraic behavior, i.e. a power law with exponent ph(‘) + 3 − D(h(‘)) which depends on the scale ‘. Taking into account the 8uctuations of the dissipative range, one has for the structure functions F ‘p if ‘D (h∗ (p)) ¡ ‘ ; Sp (‘) ∼ (B.8) ‘h(‘)p+3−D(h(‘)) if ‘D (hmin ) ¡ ‘ ¡ ‘D (h∗ (p)) : A simple calculation [83,84] shows that it is possible to
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The coding of a signal, x(t), by the exit-time approach is the following. Given a reference starting time t = t0 , one measures the
(C.1)
The limit of in
(C.2)
The link between h (j) and the j-entropy (3.45) can be obtained as follows. We note that there is a one-to-one correspondence between the (exit-time)-histories and the (j; )-histories (in the limit → 0) originating from a given j-cell. The Shannon–McMillan theorem [121] assures that the number of the typical (j; )-histories of length N , N(j; N ), is such that ln N(j; N ) h(j)N = h(j)T . For the number of typical (exit-time)-histories of lengthM , M(j; M ), we have ln M(j; M ) h (j)M . If we consider T = M t(j) , where t(j) = 1=M M i=1 ti , we must obtain the same number of (very long) histories. Therefore, from the relation M = T= t(j) we obtain
Mh (j) h (j) h (j; r ) ; = T
t(j)
t(j)
(C.3)
where the last equality is valid at least for small enough r [3]. In most of the cases, the leading j-contribution to h(j) in (C.3) is given by the mean exit time t(j) and not by h (j; r ). Anyhow, the computation of h (j; r ) is compulsory in order to recover, e.g., a zero entropy for regular (e.g. periodic) signals. One can easily estimate an upper and a lower bound for h(j) which can be computed in the exit-time scheme [3]. We use the following notation: for given j and r , h (j; r ) ≡ h ({Ki ; ki }), and we indicate with h ({ki }) and h ({Ki }) respectively the Shannon entropy of the sequence {ki } and {Ki }. By applying standard results of information theory [201] one obtains the inequalities (see [3,4] for more details): h ({ki }) 6 h ({Ki ; ki }) 6 h ({Ki }) + h ({ki }) :
(C.4)
Moreover, h ({Ki }) 6 H1 ({Ki }), where H1 ({Ki }) is the one-symbol entropy of {Ki } (i.e. the entropy of the probability distribution of the exit times measured on the scale r ) which
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can be written as
t(j) ; H1 ({Ki }) = c(j) + ln r where c(j)= − p(z) ln p(z) d z, and p(z) is the probability distribution function of the rescaled exit time z(j) = t(j)= t(j) . Finally, using the previous relations, one obtains the following bounds for the j-entropy: h ({ki }) h ({ki }) + c(j) + ln( t(j) =r ) 6 h(j) 6 : (C.5)
t(j)
t(j) Note that such bounds are relatively easy to compute and give a good estimate of h(j). In particular, as far as the scaling behavior of h(j) is concerned, the exit-time method allows for very eJcient and good estimates of the scaling exponent. The reason is that at <xed j,
t(j) automatically selects the typical time at that scale. Consequently, it is not necessary to reach very large block sizes—at least if j is not too small. So that the leading contribution is given by t(j) , and h (j; r ) introduces, at worst, a sub-leading logarithmic contribution h (j; r ) ∼ ln( t(j) =r ) (see Eq. (C.5)). In Refs. [3,4] one can found the details of the derivation and some applications.
Appendix D. Synthetic signals for turbulence In this appendix we recall some recently introduced methods for generating multi-aJne stochastic signals [27,29], whose scaling properties are fully under control. The
where we have a set of reference scales ln =2−n and ’(x) is a wavelet-like function [77], i.e. of zero mean and rapidly decaying in both real space and Fourier-space. The signal v(x) is built in
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terms of a superposition of 8uctuations, ’((x − x n; k )= ln ) of characteristic width ln and centered in di#erent points of [0; 1], x n; k = (2k + 1)=2n+1 . In [29] it has been proved that provided the coeJcients an; k are chosen by a random multiplicative process, i.e. the daughter is given in terms of the mother by a random process, an+1; k = Xan; k with X a random number identical, independent distributed for any {n; k }, then the result of the superposition is a multi-aJne function with given scaling exponents, namely T|v(x + R) − v(x)|p U ∼ R F(p) ;
with F(p) = −p=2 − log2 X p and lN 6 R 6 1. In this appendix, · indicates the average over the probability distribution of the multiplicative process. Besides the rigorous proof, the rationale for the previous result is simply that due to the hierarchical organization of the 8uctuations, one may easily see that the term dominating the expression of a velocity 8uctuation at scale R, in (D.1) is given by the couple of indices {n; k } such that n ∼ log2 (R) and x ∼ x n; k , i.e. v(x + R) − v(x) ∼ an; k . The generalization (D.1) to d-dimension is given by d(n−1) N 2 x − xn; k v(x) = an; k ’ ; ln n=1 k=1
where now the coeJcients {an; k } are given in terms of a d-dimensional dyadic multiplicative process. On the other hand, as previously said, sequential algorithms look more suitable for mimicking temporal 8uctuations. Let us now discuss how to construct these stochastic multi-aJne <elds. With the application to time 8uctuations in mind, we will denote now the stochastic 1-dimensional functions with u(t). The signal u(t) is obtained by a superposition of functions with di#erent characteristic times, representing eddies of various sizes [29]: u(t) =
N
un (t) :
(D.2)
n=1
The functions un (t) are de
(D.3)
where the gn (t) are independent stationary random processes, whose correlation times are supposed to be n = (ln )= , where = = 1 − h (i.e. n are the eddy-turn-over time at scale ln ) in the quasi-Lagrangian frame (Ref. [152]) and = = 1 if one considers u(t) as the time signal in a given point, and gn2 = (ln )2h , where h is the HVolder exponent. For a signal mimicking a turbulent 8ow, ignoring intermittency, we would have h = 13 . Scaling will appear for all time delays larger than the UV cuto# N and smaller than the IR cuto# 1 . The xj (t) are independent, positive de
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The following arguments show that the process de
where the temporal dependency of an; k (t) is chosen following the sequential algorithm while its spatial part are given by the dyadic structure of the non-sequential algorithm. In (D.6) we have used the notation vL (x; t) in order to stress the typical Lagrangian character of such a <eld. We are now able to guess a good candidate for the same <eld measured in the laboratoryreference frame, i.e. where the time properties are dominated by the sweeping of small scales by large scales. Indeed, in order to reproduce the sweeping e#ects one needs that the centers {xn; k } of the wavelets-like functions move according a swept dynamics. To do so, let us de
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where the di#erence with the previous de
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[15] P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38 (1988) 364. [16] F. Bagnoli, R. Rechtman, S. Ru#o, Damage spreading and Lyapunov exponent in cellular automata, Phys. Lett. A 172 (1992) 34. [17] F. Bagnoli, R. Rechtman, Synchronization and maximum Lyapunov exponent in cellular automata, Phys. Rev. E 59 (1999) R1307. [18] C. Basdevant, B. Legras, R. Sadourny, M. Beland, A study of barotropic model 8ows: intermittency, waves and predictability, J. Atmos. Sci. 38 (1981) 2305. [19] G.K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1953. [20] G. Benettin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them, Meccanica 15 (1980) 9. [21] C.H. Bennet, How to de
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CONTENTS VOLUME 356 Y. Kraftmakher. Modulation calorimetry and related techniques
1
V.M. Shabaev. Two-time Green's function method in quantum electrodynamics of high-Z few-electron atoms
119
J.A. Adam. The mathematical physics of rainbows and glories
229
G. Bo!etta, M. Cencini, M. Falcioni, A. Vulpiani. Predictability: a way to characterize complexity
367
Contents of volume
475
Forthcoming issues
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FORTHCOMING ISSUES A. Wacker. Semiconductor superlattices: a model system for nonlinear transport I.L. Shapiro. Physical aspects of the space}time torsion M.J. Brunger, S.J. Buckman. Electron}molecule scattering cross sections. I. Experimental techniques and data for diatomic molecules S.Y. Wu, C.S. Jayanthi. Order-N methodologies and their applications V. Barone, A. Drago, P. Ratcli!e. Transverse polarisation of quarks in hadrons M. Baer. Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems S.-T. Hong, Y.-J. Park. Static properties of chiral models with SU(3) group structure W.M. Alberico, S.M. Bilenky, C. Maieron. Strangeness in the nucleon: neutrino}nucleon and polarized electron}nucleon scattering M. Bianchetti, P.F. Buonsante, F. Ginelli, H.E. Roman, R.A. Broglia, F. Alasia. Ab-initio study of the electronic response and polarizability of carbon chains C.M. Varma, Z. Nussinov, W. van Saarloos. Singular Fermi liquids J.-P. Blaizot, E. Iancu. The quark}gluon plasma: collective dynamics and hard thermal loops A. Sopczak. Higgs physics at LEP-1 I.L. Aleiner, P.W. Brouwer, L.I. Glazman. Quantum e!ect in Coulomb blockade A. Altland, B.D. Simons, M. Zirnbauer. Theories of low-energy quasi-particle states in disordered d-wave superconductors J.A. Krommes. Fundamental descriptions of plasma turbulence in magnetic "elds J.D. Vergados. The neutrinoless double beta decay from a modern perspective C.-I. Um, K.-H. Yeon, T.F. George. The quantum damped harmonic oscillator
PII: S0370-1573(01)00086-2
